diff --git a/src/ZF/ZF_Base.thy b/src/ZF/ZF_Base.thy
--- a/src/ZF/ZF_Base.thy
+++ b/src/ZF/ZF_Base.thy
@@ -1,650 +1,650 @@
 (*  Title:      ZF/ZF_Base.thy
     Author:     Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
     Copyright   1993  University of Cambridge
 *)
 
 section \<open>Base of Zermelo-Fraenkel Set Theory\<close>
 
 theory ZF_Base
 imports FOL
 begin
 
 subsection \<open>Signature\<close>
 
 declare [[eta_contract = false]]
 
 typedecl i
 instance i :: "term" ..
 
 axiomatization mem :: "[i, i] \<Rightarrow> o"  (infixl \<open>\<in>\<close> 50)  \<comment> \<open>membership relation\<close>
   and zero :: "i"  (\<open>0\<close>)  \<comment> \<open>the empty set\<close>
   and Pow :: "i \<Rightarrow> i"  \<comment> \<open>power sets\<close>
   and Inf :: "i"  \<comment> \<open>infinite set\<close>
   and Union :: "i \<Rightarrow> i"  (\<open>\<Union>_\<close> [90] 90)
   and PrimReplace :: "[i, [i, i] \<Rightarrow> o] \<Rightarrow> i"
 
 abbreviation not_mem :: "[i, i] \<Rightarrow> o"  (infixl \<open>\<notin>\<close> 50)  \<comment> \<open>negated membership relation\<close>
   where "x \<notin> y \<equiv> \<not> (x \<in> y)"
 
 
 subsection \<open>Bounded Quantifiers\<close>
 
 definition Ball :: "[i, i \<Rightarrow> o] \<Rightarrow> o"
   where "Ball(A, P) \<equiv> \<forall>x. x\<in>A \<longrightarrow> P(x)"
 
 definition Bex :: "[i, i \<Rightarrow> o] \<Rightarrow> o"
   where "Bex(A, P) \<equiv> \<exists>x. x\<in>A \<and> P(x)"
 
 syntax
   "_Ball" :: "[pttrn, i, o] \<Rightarrow> o"  (\<open>(3\<forall>_\<in>_./ _)\<close> 10)
   "_Bex" :: "[pttrn, i, o] \<Rightarrow> o"  (\<open>(3\<exists>_\<in>_./ _)\<close> 10)
 translations
   "\<forall>x\<in>A. P" \<rightleftharpoons> "CONST Ball(A, \<lambda>x. P)"
   "\<exists>x\<in>A. P" \<rightleftharpoons> "CONST Bex(A, \<lambda>x. P)"
 
 
 subsection \<open>Variations on Replacement\<close>
 
 (* Derived form of replacement, restricting P to its functional part.
    The resulting set (for functional P) is the same as with
    PrimReplace, but the rules are simpler. *)
 definition Replace :: "[i, [i, i] \<Rightarrow> o] \<Rightarrow> i"
   where "Replace(A,P) == PrimReplace(A, %x y. (\<exists>!z. P(x,z)) & P(x,y))"
 
 syntax
   "_Replace"  :: "[pttrn, pttrn, i, o] => i"  (\<open>(1{_ ./ _ \<in> _, _})\<close>)
 translations
   "{y. x\<in>A, Q}" \<rightleftharpoons> "CONST Replace(A, \<lambda>x y. Q)"
 
 
 (* Functional form of replacement -- analgous to ML's map functional *)
 
 definition RepFun :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
   where "RepFun(A,f) == {y . x\<in>A, y=f(x)}"
 
 syntax
   "_RepFun" :: "[i, pttrn, i] => i"  (\<open>(1{_ ./ _ \<in> _})\<close> [51,0,51])
 translations
   "{b. x\<in>A}" \<rightleftharpoons> "CONST RepFun(A, \<lambda>x. b)"
 
 
 (* Separation and Pairing can be derived from the Replacement
    and Powerset Axioms using the following definitions. *)
 definition Collect :: "[i, i \<Rightarrow> o] \<Rightarrow> i"
   where "Collect(A,P) == {y . x\<in>A, x=y & P(x)}"
 
 syntax
   "_Collect" :: "[pttrn, i, o] \<Rightarrow> i"  (\<open>(1{_ \<in> _ ./ _})\<close>)
 translations
   "{x\<in>A. P}" \<rightleftharpoons> "CONST Collect(A, \<lambda>x. P)"
 
 
 subsection \<open>General union and intersection\<close>
 
 definition Inter :: "i => i"  (\<open>\<Inter>_\<close> [90] 90)
   where "\<Inter>(A) == { x\<in>\<Union>(A) . \<forall>y\<in>A. x\<in>y}"
 
 syntax
   "_UNION" :: "[pttrn, i, i] => i"  (\<open>(3\<Union>_\<in>_./ _)\<close> 10)
   "_INTER" :: "[pttrn, i, i] => i"  (\<open>(3\<Inter>_\<in>_./ _)\<close> 10)
 translations
   "\<Union>x\<in>A. B" == "CONST Union({B. x\<in>A})"
   "\<Inter>x\<in>A. B" == "CONST Inter({B. x\<in>A})"
 
 
 subsection \<open>Finite sets and binary operations\<close>
 
 (*Unordered pairs (Upair) express binary union/intersection and cons;
   set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...)*)
 
 definition Upair :: "[i, i] => i"
   where "Upair(a,b) == {y. x\<in>Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}"
 
 definition Subset :: "[i, i] \<Rightarrow> o"  (infixl \<open>\<subseteq>\<close> 50)  \<comment> \<open>subset relation\<close>
   where subset_def: "A \<subseteq> B \<equiv> \<forall>x\<in>A. x\<in>B"
 
 definition Diff :: "[i, i] \<Rightarrow> i"  (infixl \<open>-\<close> 65)  \<comment> \<open>set difference\<close>
   where "A - B == { x\<in>A . ~(x\<in>B) }"
 
 definition Un :: "[i, i] \<Rightarrow> i"  (infixl \<open>\<union>\<close> 65)  \<comment> \<open>binary union\<close>
   where "A \<union> B == \<Union>(Upair(A,B))"
 
 definition Int :: "[i, i] \<Rightarrow> i"  (infixl \<open>\<inter>\<close> 70)  \<comment> \<open>binary intersection\<close>
   where "A \<inter> B == \<Inter>(Upair(A,B))"
 
 definition cons :: "[i, i] => i"
   where "cons(a,A) == Upair(a,a) \<union> A"
 
 definition succ :: "i => i"
   where "succ(i) == cons(i, i)"
 
 nonterminal "is"
 syntax
   "" :: "i \<Rightarrow> is"  (\<open>_\<close>)
   "_Enum" :: "[i, is] \<Rightarrow> is"  (\<open>_,/ _\<close>)
   "_Finset" :: "is \<Rightarrow> i"  (\<open>{(_)}\<close>)
 translations
   "{x, xs}" == "CONST cons(x, {xs})"
   "{x}" == "CONST cons(x, 0)"
 
 
 subsection \<open>Axioms\<close>
 
 (* ZF axioms -- see Suppes p.238
    Axioms for Union, Pow and Replace state existence only,
    uniqueness is derivable using extensionality. *)
 
 axiomatization
 where
   extension:     "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A" and
   Union_iff:     "A \<in> \<Union>(C) \<longleftrightarrow> (\<exists>B\<in>C. A\<in>B)" and
   Pow_iff:       "A \<in> Pow(B) \<longleftrightarrow> A \<subseteq> B" and
 
   (*We may name this set, though it is not uniquely defined.*)
   infinity:      "0 \<in> Inf \<and> (\<forall>y\<in>Inf. succ(y) \<in> Inf)" and
 
   (*This formulation facilitates case analysis on A.*)
   foundation:    "A = 0 \<or> (\<exists>x\<in>A. \<forall>y\<in>x. y\<notin>A)" and
 
   (*Schema axiom since predicate P is a higher-order variable*)
   replacement:   "(\<forall>x\<in>A. \<forall>y z. P(x,y) \<and> P(x,z) \<longrightarrow> y = z) \<Longrightarrow>
                          b \<in> PrimReplace(A,P) \<longleftrightarrow> (\<exists>x\<in>A. P(x,b))"
 
 
 subsection \<open>Definite descriptions -- via Replace over the set "1"\<close>
 
 definition The :: "(i \<Rightarrow> o) \<Rightarrow> i"  (binder \<open>THE \<close> 10)
   where the_def: "The(P)    == \<Union>({y . x \<in> {0}, P(y)})"
 
 definition If :: "[o, i, i] \<Rightarrow> i"  (\<open>(if (_)/ then (_)/ else (_))\<close> [10] 10)
   where if_def: "if P then a else b == THE z. P & z=a | ~P & z=b"
 
 abbreviation (input)
   old_if :: "[o, i, i] => i"  (\<open>if '(_,_,_')\<close>)
   where "if(P,a,b) == If(P,a,b)"
 
 
 subsection \<open>Ordered Pairing\<close>
 
 (* this "symmetric" definition works better than {{a}, {a,b}} *)
 definition Pair :: "[i, i] => i"
   where "Pair(a,b) == {{a,a}, {a,b}}"
 
 definition fst :: "i \<Rightarrow> i"
   where "fst(p) == THE a. \<exists>b. p = Pair(a, b)"
 
 definition snd :: "i \<Rightarrow> i"
   where "snd(p) == THE b. \<exists>a. p = Pair(a, b)"
 
 definition split :: "[[i, i] \<Rightarrow> 'a, i] \<Rightarrow> 'a::{}"  \<comment> \<open>for pattern-matching\<close>
   where "split(c) == \<lambda>p. c(fst(p), snd(p))"
 
 (* Patterns -- extends pre-defined type "pttrn" used in abstractions *)
 nonterminal patterns
 syntax
   "_pattern"  :: "patterns => pttrn"         (\<open>\<langle>_\<rangle>\<close>)
   ""          :: "pttrn => patterns"         (\<open>_\<close>)
   "_patterns" :: "[pttrn, patterns] => patterns"  (\<open>_,/_\<close>)
   "_Tuple"    :: "[i, is] => i"              (\<open>\<langle>(_,/ _)\<rangle>\<close>)
 translations
   "\<langle>x, y, z\<rangle>"   == "\<langle>x, \<langle>y, z\<rangle>\<rangle>"
   "\<langle>x, y\<rangle>"      == "CONST Pair(x, y)"
   "\<lambda>\<langle>x,y,zs\<rangle>.b" == "CONST split(\<lambda>x \<langle>y,zs\<rangle>.b)"
   "\<lambda>\<langle>x,y\<rangle>.b"    == "CONST split(\<lambda>x y. b)"
 
 definition Sigma :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
   where "Sigma(A,B) == \<Union>x\<in>A. \<Union>y\<in>B(x). {\<langle>x,y\<rangle>}"
 
 abbreviation cart_prod :: "[i, i] => i"  (infixr \<open>\<times>\<close> 80)  \<comment> \<open>Cartesian product\<close>
   where "A \<times> B \<equiv> Sigma(A, \<lambda>_. B)"
 
 
 subsection \<open>Relations and Functions\<close>
 
 (*converse of relation r, inverse of function*)
 definition converse :: "i \<Rightarrow> i"
   where "converse(r) == {z. w\<in>r, \<exists>x y. w=\<langle>x,y\<rangle> \<and> z=\<langle>y,x\<rangle>}"
 
 definition domain :: "i \<Rightarrow> i"
   where "domain(r) == {x. w\<in>r, \<exists>y. w=\<langle>x,y\<rangle>}"
 
 definition range :: "i \<Rightarrow> i"
   where "range(r) == domain(converse(r))"
 
 definition field :: "i \<Rightarrow> i"
   where "field(r) == domain(r) \<union> range(r)"
 
 definition relation :: "i \<Rightarrow> o"  \<comment> \<open>recognizes sets of pairs\<close>
   where "relation(r) == \<forall>z\<in>r. \<exists>x y. z = \<langle>x,y\<rangle>"
 
 definition "function" :: "i \<Rightarrow> o"  \<comment> \<open>recognizes functions; can have non-pairs\<close>
   where "function(r) == \<forall>x y. \<langle>x,y\<rangle> \<in> r \<longrightarrow> (\<forall>y'. \<langle>x,y'\<rangle> \<in> r \<longrightarrow> y = y')"
 
 definition Image :: "[i, i] \<Rightarrow> i"  (infixl \<open>``\<close> 90)  \<comment> \<open>image\<close>
   where image_def: "r `` A  == {y \<in> range(r). \<exists>x\<in>A. \<langle>x,y\<rangle> \<in> r}"
 
 definition vimage :: "[i, i] \<Rightarrow> i"  (infixl \<open>-``\<close> 90)  \<comment> \<open>inverse image\<close>
   where vimage_def: "r -`` A == converse(r)``A"
 
 (* Restrict the relation r to the domain A *)
 definition restrict :: "[i, i] \<Rightarrow> i"
   where "restrict(r,A) == {z \<in> r. \<exists>x\<in>A. \<exists>y. z = \<langle>x,y\<rangle>}"
 
 
 (* Abstraction, application and Cartesian product of a family of sets *)
 
 definition Lambda :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
   where lam_def: "Lambda(A,b) == {\<langle>x,b(x)\<rangle>. x\<in>A}"
 
 definition "apply" :: "[i, i] \<Rightarrow> i"  (infixl \<open>`\<close> 90)  \<comment> \<open>function application\<close>
   where "f`a == \<Union>(f``{a})"
 
 definition Pi :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
   where "Pi(A,B) == {f\<in>Pow(Sigma(A,B)). A\<subseteq>domain(f) & function(f)}"
 
-abbreviation function_space :: "[i, i] \<Rightarrow> i"  (infixr \<open>->\<close> 60)  \<comment> \<open>function space\<close>
-  where "A -> B \<equiv> Pi(A, \<lambda>_. B)"
+abbreviation function_space :: "[i, i] \<Rightarrow> i"  (infixr \<open>\<rightarrow>\<close> 60)  \<comment> \<open>function space\<close>
+  where "A \<rightarrow> B \<equiv> Pi(A, \<lambda>_. B)"
 
 
 (* binder syntax *)
 
 syntax
   "_PROD"     :: "[pttrn, i, i] => i"        (\<open>(3\<Prod>_\<in>_./ _)\<close> 10)
   "_SUM"      :: "[pttrn, i, i] => i"        (\<open>(3\<Sum>_\<in>_./ _)\<close> 10)
   "_lam"      :: "[pttrn, i, i] => i"        (\<open>(3\<lambda>_\<in>_./ _)\<close> 10)
 translations
   "\<Prod>x\<in>A. B"   == "CONST Pi(A, \<lambda>x. B)"
   "\<Sum>x\<in>A. B"   == "CONST Sigma(A, \<lambda>x. B)"
   "\<lambda>x\<in>A. f"    == "CONST Lambda(A, \<lambda>x. f)"
 
 
 subsection \<open>ASCII syntax\<close>
 
 notation (ASCII)
   cart_prod       (infixr \<open>*\<close> 80) and
   Int             (infixl \<open>Int\<close> 70) and
   Un              (infixl \<open>Un\<close> 65) and
-  function_space  (infixr \<open>\<rightarrow>\<close> 60) and
+  function_space  (infixr \<open>->\<close> 60) and
   Subset          (infixl \<open><=\<close> 50) and
   mem             (infixl \<open>:\<close> 50) and
   not_mem         (infixl \<open>~:\<close> 50)
 
 syntax (ASCII)
   "_Ball"     :: "[pttrn, i, o] => o"        (\<open>(3ALL _:_./ _)\<close> 10)
   "_Bex"      :: "[pttrn, i, o] => o"        (\<open>(3EX _:_./ _)\<close> 10)
   "_Collect"  :: "[pttrn, i, o] => i"        (\<open>(1{_: _ ./ _})\<close>)
   "_Replace"  :: "[pttrn, pttrn, i, o] => i" (\<open>(1{_ ./ _: _, _})\<close>)
   "_RepFun"   :: "[i, pttrn, i] => i"        (\<open>(1{_ ./ _: _})\<close> [51,0,51])
   "_UNION"    :: "[pttrn, i, i] => i"        (\<open>(3UN _:_./ _)\<close> 10)
   "_INTER"    :: "[pttrn, i, i] => i"        (\<open>(3INT _:_./ _)\<close> 10)
   "_PROD"     :: "[pttrn, i, i] => i"        (\<open>(3PROD _:_./ _)\<close> 10)
   "_SUM"      :: "[pttrn, i, i] => i"        (\<open>(3SUM _:_./ _)\<close> 10)
   "_lam"      :: "[pttrn, i, i] => i"        (\<open>(3lam _:_./ _)\<close> 10)
   "_Tuple"    :: "[i, is] => i"              (\<open><(_,/ _)>\<close>)
   "_pattern"  :: "patterns => pttrn"         (\<open><_>\<close>)
 
 
 subsection \<open>Substitution\<close>
 
 (*Useful examples:  singletonI RS subst_elem,  subst_elem RSN (2,IntI) *)
 lemma subst_elem: "[| b\<in>A;  a=b |] ==> a\<in>A"
 by (erule ssubst, assumption)
 
 
 subsection\<open>Bounded universal quantifier\<close>
 
 lemma ballI [intro!]: "[| !!x. x\<in>A ==> P(x) |] ==> \<forall>x\<in>A. P(x)"
 by (simp add: Ball_def)
 
 lemmas strip = impI allI ballI
 
 lemma bspec [dest?]: "[| \<forall>x\<in>A. P(x);  x: A |] ==> P(x)"
 by (simp add: Ball_def)
 
 (*Instantiates x first: better for automatic theorem proving?*)
 lemma rev_ballE [elim]:
     "[| \<forall>x\<in>A. P(x);  x\<notin>A ==> Q;  P(x) ==> Q |] ==> Q"
 by (simp add: Ball_def, blast)
 
 lemma ballE: "[| \<forall>x\<in>A. P(x);  P(x) ==> Q;  x\<notin>A ==> Q |] ==> Q"
 by blast
 
 (*Used in the datatype package*)
 lemma rev_bspec: "[| x: A;  \<forall>x\<in>A. P(x) |] ==> P(x)"
 by (simp add: Ball_def)
 
 (*Trival rewrite rule;   @{term"(\<forall>x\<in>A.P)<->P"} holds only if A is nonempty!*)
 lemma ball_triv [simp]: "(\<forall>x\<in>A. P) <-> ((\<exists>x. x\<in>A) \<longrightarrow> P)"
 by (simp add: Ball_def)
 
 (*Congruence rule for rewriting*)
 lemma ball_cong [cong]:
     "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |] ==> (\<forall>x\<in>A. P(x)) <-> (\<forall>x\<in>A'. P'(x))"
 by (simp add: Ball_def)
 
 lemma atomize_ball:
     "(!!x. x \<in> A ==> P(x)) == Trueprop (\<forall>x\<in>A. P(x))"
   by (simp only: Ball_def atomize_all atomize_imp)
 
 lemmas [symmetric, rulify] = atomize_ball
   and [symmetric, defn] = atomize_ball
 
 
 subsection\<open>Bounded existential quantifier\<close>
 
 lemma bexI [intro]: "[| P(x);  x: A |] ==> \<exists>x\<in>A. P(x)"
 by (simp add: Bex_def, blast)
 
 (*The best argument order when there is only one @{term"x\<in>A"}*)
 lemma rev_bexI: "[| x\<in>A;  P(x) |] ==> \<exists>x\<in>A. P(x)"
 by blast
 
 (*Not of the general form for such rules. The existential quanitifer becomes universal. *)
 lemma bexCI: "[| \<forall>x\<in>A. ~P(x) ==> P(a);  a: A |] ==> \<exists>x\<in>A. P(x)"
 by blast
 
 lemma bexE [elim!]: "[| \<exists>x\<in>A. P(x);  !!x. [| x\<in>A; P(x) |] ==> Q |] ==> Q"
 by (simp add: Bex_def, blast)
 
 (*We do not even have @{term"(\<exists>x\<in>A. True) <-> True"} unless @{term"A" is nonempty!!*)
 lemma bex_triv [simp]: "(\<exists>x\<in>A. P) <-> ((\<exists>x. x\<in>A) & P)"
 by (simp add: Bex_def)
 
 lemma bex_cong [cong]:
     "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |]
      ==> (\<exists>x\<in>A. P(x)) <-> (\<exists>x\<in>A'. P'(x))"
 by (simp add: Bex_def cong: conj_cong)
 
 
 
 subsection\<open>Rules for subsets\<close>
 
 lemma subsetI [intro!]:
     "(!!x. x\<in>A ==> x\<in>B) ==> A \<subseteq> B"
 by (simp add: subset_def)
 
 (*Rule in Modus Ponens style [was called subsetE] *)
 lemma subsetD [elim]: "[| A \<subseteq> B;  c\<in>A |] ==> c\<in>B"
 apply (unfold subset_def)
 apply (erule bspec, assumption)
 done
 
 (*Classical elimination rule*)
 lemma subsetCE [elim]:
     "[| A \<subseteq> B;  c\<notin>A ==> P;  c\<in>B ==> P |] ==> P"
 by (simp add: subset_def, blast)
 
 (*Sometimes useful with premises in this order*)
 lemma rev_subsetD: "[| c\<in>A; A<=B |] ==> c\<in>B"
 by blast
 
 lemma contra_subsetD: "[| A \<subseteq> B; c \<notin> B |] ==> c \<notin> A"
 by blast
 
 lemma rev_contra_subsetD: "[| c \<notin> B;  A \<subseteq> B |] ==> c \<notin> A"
 by blast
 
 lemma subset_refl [simp]: "A \<subseteq> A"
 by blast
 
 lemma subset_trans: "[| A<=B;  B<=C |] ==> A<=C"
 by blast
 
 (*Useful for proving A<=B by rewriting in some cases*)
 lemma subset_iff:
      "A<=B <-> (\<forall>x. x\<in>A \<longrightarrow> x\<in>B)"
 apply (unfold subset_def Ball_def)
 apply (rule iff_refl)
 done
 
 text\<open>For calculations\<close>
 declare subsetD [trans] rev_subsetD [trans] subset_trans [trans]
 
 
 subsection\<open>Rules for equality\<close>
 
 (*Anti-symmetry of the subset relation*)
 lemma equalityI [intro]: "[| A \<subseteq> B;  B \<subseteq> A |] ==> A = B"
 by (rule extension [THEN iffD2], rule conjI)
 
 
 lemma equality_iffI: "(!!x. x\<in>A <-> x\<in>B) ==> A = B"
 by (rule equalityI, blast+)
 
 lemmas equalityD1 = extension [THEN iffD1, THEN conjunct1]
 lemmas equalityD2 = extension [THEN iffD1, THEN conjunct2]
 
 lemma equalityE: "[| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P"
 by (blast dest: equalityD1 equalityD2)
 
 lemma equalityCE:
     "[| A = B;  [| c\<in>A; c\<in>B |] ==> P;  [| c\<notin>A; c\<notin>B |] ==> P |]  ==>  P"
 by (erule equalityE, blast)
 
 lemma equality_iffD:
   "A = B ==> (!!x. x \<in> A <-> x \<in> B)"
   by auto
 
 
 subsection\<open>Rules for Replace -- the derived form of replacement\<close>
 
 lemma Replace_iff:
     "b \<in> {y. x\<in>A, P(x,y)}  <->  (\<exists>x\<in>A. P(x,b) & (\<forall>y. P(x,y) \<longrightarrow> y=b))"
 apply (unfold Replace_def)
 apply (rule replacement [THEN iff_trans], blast+)
 done
 
 (*Introduction; there must be a unique y such that P(x,y), namely y=b. *)
 lemma ReplaceI [intro]:
     "[| P(x,b);  x: A;  !!y. P(x,y) ==> y=b |] ==>
      b \<in> {y. x\<in>A, P(x,y)}"
 by (rule Replace_iff [THEN iffD2], blast)
 
 (*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *)
 lemma ReplaceE:
     "[| b \<in> {y. x\<in>A, P(x,y)};
         !!x. [| x: A;  P(x,b);  \<forall>y. P(x,y)\<longrightarrow>y=b |] ==> R
      |] ==> R"
 by (rule Replace_iff [THEN iffD1, THEN bexE], simp+)
 
 (*As above but without the (generally useless) 3rd assumption*)
 lemma ReplaceE2 [elim!]:
     "[| b \<in> {y. x\<in>A, P(x,y)};
         !!x. [| x: A;  P(x,b) |] ==> R
      |] ==> R"
 by (erule ReplaceE, blast)
 
 lemma Replace_cong [cong]:
     "[| A=B;  !!x y. x\<in>B ==> P(x,y) <-> Q(x,y) |] ==>
      Replace(A,P) = Replace(B,Q)"
 apply (rule equality_iffI)
 apply (simp add: Replace_iff)
 done
 
 
 subsection\<open>Rules for RepFun\<close>
 
 lemma RepFunI: "a \<in> A ==> f(a) \<in> {f(x). x\<in>A}"
 by (simp add: RepFun_def Replace_iff, blast)
 
 (*Useful for coinduction proofs*)
 lemma RepFun_eqI [intro]: "[| b=f(a);  a \<in> A |] ==> b \<in> {f(x). x\<in>A}"
 apply (erule ssubst)
 apply (erule RepFunI)
 done
 
 lemma RepFunE [elim!]:
     "[| b \<in> {f(x). x\<in>A};
         !!x.[| x\<in>A;  b=f(x) |] ==> P |] ==>
      P"
 by (simp add: RepFun_def Replace_iff, blast)
 
 lemma RepFun_cong [cong]:
     "[| A=B;  !!x. x\<in>B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
 by (simp add: RepFun_def)
 
 lemma RepFun_iff [simp]: "b \<in> {f(x). x\<in>A} <-> (\<exists>x\<in>A. b=f(x))"
 by (unfold Bex_def, blast)
 
 lemma triv_RepFun [simp]: "{x. x\<in>A} = A"
 by blast
 
 
 subsection\<open>Rules for Collect -- forming a subset by separation\<close>
 
 (*Separation is derivable from Replacement*)
 lemma separation [simp]: "a \<in> {x\<in>A. P(x)} <-> a\<in>A & P(a)"
 by (unfold Collect_def, blast)
 
 lemma CollectI [intro!]: "[| a\<in>A;  P(a) |] ==> a \<in> {x\<in>A. P(x)}"
 by simp
 
 lemma CollectE [elim!]: "[| a \<in> {x\<in>A. P(x)};  [| a\<in>A; P(a) |] ==> R |] ==> R"
 by simp
 
 lemma CollectD1: "a \<in> {x\<in>A. P(x)} ==> a\<in>A"
 by (erule CollectE, assumption)
 
 lemma CollectD2: "a \<in> {x\<in>A. P(x)} ==> P(a)"
 by (erule CollectE, assumption)
 
 lemma Collect_cong [cong]:
     "[| A=B;  !!x. x\<in>B ==> P(x) <-> Q(x) |]
      ==> Collect(A, %x. P(x)) = Collect(B, %x. Q(x))"
 by (simp add: Collect_def)
 
 
 subsection\<open>Rules for Unions\<close>
 
 declare Union_iff [simp]
 
 (*The order of the premises presupposes that C is rigid; A may be flexible*)
 lemma UnionI [intro]: "[| B: C;  A: B |] ==> A: \<Union>(C)"
 by (simp, blast)
 
 lemma UnionE [elim!]: "[| A \<in> \<Union>(C);  !!B.[| A: B;  B: C |] ==> R |] ==> R"
 by (simp, blast)
 
 
 subsection\<open>Rules for Unions of families\<close>
 (* @{term"\<Union>x\<in>A. B(x)"} abbreviates @{term"\<Union>({B(x). x\<in>A})"} *)
 
 lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B(x)) <-> (\<exists>x\<in>A. b \<in> B(x))"
 by (simp add: Bex_def, blast)
 
 (*The order of the premises presupposes that A is rigid; b may be flexible*)
 lemma UN_I: "[| a: A;  b: B(a) |] ==> b: (\<Union>x\<in>A. B(x))"
 by (simp, blast)
 
 
 lemma UN_E [elim!]:
     "[| b \<in> (\<Union>x\<in>A. B(x));  !!x.[| x: A;  b: B(x) |] ==> R |] ==> R"
 by blast
 
 lemma UN_cong:
     "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Union>x\<in>A. C(x)) = (\<Union>x\<in>B. D(x))"
 by simp
 
 
 (*No "Addcongs [UN_cong]" because @{term\<Union>} is a combination of constants*)
 
 (* UN_E appears before UnionE so that it is tried first, to avoid expensive
   calls to hyp_subst_tac.  Cannot include UN_I as it is unsafe: would enlarge
   the search space.*)
 
 
 subsection\<open>Rules for the empty set\<close>
 
 (*The set @{term"{x\<in>0. False}"} is empty; by foundation it equals 0
   See Suppes, page 21.*)
 lemma not_mem_empty [simp]: "a \<notin> 0"
 apply (cut_tac foundation)
 apply (best dest: equalityD2)
 done
 
 lemmas emptyE [elim!] = not_mem_empty [THEN notE]
 
 
 lemma empty_subsetI [simp]: "0 \<subseteq> A"
 by blast
 
 lemma equals0I: "[| !!y. y\<in>A ==> False |] ==> A=0"
 by blast
 
 lemma equals0D [dest]: "A=0 ==> a \<notin> A"
 by blast
 
 declare sym [THEN equals0D, dest]
 
 lemma not_emptyI: "a\<in>A ==> A \<noteq> 0"
 by blast
 
 lemma not_emptyE:  "[| A \<noteq> 0;  !!x. x\<in>A ==> R |] ==> R"
 by blast
 
 
 subsection\<open>Rules for Inter\<close>
 
 (*Not obviously useful for proving InterI, InterD, InterE*)
 lemma Inter_iff: "A \<in> \<Inter>(C) <-> (\<forall>x\<in>C. A: x) & C\<noteq>0"
 by (simp add: Inter_def Ball_def, blast)
 
 (* Intersection is well-behaved only if the family is non-empty! *)
 lemma InterI [intro!]:
     "[| !!x. x: C ==> A: x;  C\<noteq>0 |] ==> A \<in> \<Inter>(C)"
 by (simp add: Inter_iff)
 
 (*A "destruct" rule -- every B in C contains A as an element, but
   A\<in>B can hold when B\<in>C does not!  This rule is analogous to "spec". *)
 lemma InterD [elim, Pure.elim]: "[| A \<in> \<Inter>(C);  B \<in> C |] ==> A \<in> B"
 by (unfold Inter_def, blast)
 
 (*"Classical" elimination rule -- does not require exhibiting @{term"B\<in>C"} *)
 lemma InterE [elim]:
     "[| A \<in> \<Inter>(C);  B\<notin>C ==> R;  A\<in>B ==> R |] ==> R"
 by (simp add: Inter_def, blast)
 
 
 subsection\<open>Rules for Intersections of families\<close>
 
 (* @{term"\<Inter>x\<in>A. B(x)"} abbreviates @{term"\<Inter>({B(x). x\<in>A})"} *)
 
 lemma INT_iff: "b \<in> (\<Inter>x\<in>A. B(x)) <-> (\<forall>x\<in>A. b \<in> B(x)) & A\<noteq>0"
 by (force simp add: Inter_def)
 
 lemma INT_I: "[| !!x. x: A ==> b: B(x);  A\<noteq>0 |] ==> b: (\<Inter>x\<in>A. B(x))"
 by blast
 
 lemma INT_E: "[| b \<in> (\<Inter>x\<in>A. B(x));  a: A |] ==> b \<in> B(a)"
 by blast
 
 lemma INT_cong:
     "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Inter>x\<in>A. C(x)) = (\<Inter>x\<in>B. D(x))"
 by simp
 
 (*No "Addcongs [INT_cong]" because @{term\<Inter>} is a combination of constants*)
 
 
 subsection\<open>Rules for Powersets\<close>
 
 lemma PowI: "A \<subseteq> B ==> A \<in> Pow(B)"
 by (erule Pow_iff [THEN iffD2])
 
 lemma PowD: "A \<in> Pow(B)  ==>  A<=B"
 by (erule Pow_iff [THEN iffD1])
 
 declare Pow_iff [iff]
 
 lemmas Pow_bottom = empty_subsetI [THEN PowI]    \<comment> \<open>\<^term>\<open>0 \<in> Pow(B)\<close>\<close>
 lemmas Pow_top = subset_refl [THEN PowI]         \<comment> \<open>\<^term>\<open>A \<in> Pow(A)\<close>\<close>
 
 
 subsection\<open>Cantor's Theorem: There is no surjection from a set to its powerset.\<close>
 
 (*The search is undirected.  Allowing redundant introduction rules may
   make it diverge.  Variable b represents ANY map, such as
   (lam x\<in>A.b(x)): A->Pow(A). *)
 lemma cantor: "\<exists>S \<in> Pow(A). \<forall>x\<in>A. b(x) \<noteq> S"
 by (best elim!: equalityCE del: ReplaceI RepFun_eqI)
 
 end