mmtheorems45 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4401-4500 - Page 45 of 123

Type	Label	Description
Statement
Definition	df-er 4401	Define the equivalence predicate. R need not be a relation but ordinarily will be, in which case we call it an equivalence relation. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. Some definitions in the literature may also require that R be a relation. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 4402 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 4415, ersymb 4413, and ertr 4414.
|- (Er R <-> (`'R u. (R o. R)) (_ R)
Theorem	dfer2 4402	Alternate definition of equivalence predicate.
|- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
Definition	df-ec 4403	Define the R -coset of A. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of A modulo R when R is an equivalence relation (i.e. when Er R; see dfer2 4402). In this case, A is a representative (member) of the equivalence class [A]R, which contains all sets that are equivalent to A. Definition of [Enderton] p. 57 uses the notation [A] (subscript) R, although we simply follow the brackets by R since we don't have subscripted expressions. For an alternate definition, see dfec2 4404.
|- [A]R = (R"{A})
Theorem	dfec2 4404	Alternate definition of R-coset of A. Definition 34 of [Suppes] p. 81.
|- A e. V   =>   |- [A]R = {y | ARy}
Theorem	ecexg 4405	An equivalence class modulo a set is a set.
|- (R e. B -> [A]R e. V)
Definition	df-qs 4406	Define quotient set. R is usually an equivalence relation. Definition of [Enderton] p. 58.
|- (A/.R) = {y | E.x e. A y = [x]R}
Theorem	ereq 4407	Equality theorem for equivalence predicate.
|- (R = S -> (Er R <-> Er S))
Theorem	ster 4408	A symmetric, transitive relation is an equivalence relation.
|- (xRy -> yRx)   &   |- ((xRy /\ yRz) -> xRz)   =>   |- Er R
Theorem	ider 4409	The identity relation is an equivalence relation.
|- Er I
Theorem	eqerlem 4410	Lemma for eqer 4411.
Theorem	eqer 4411	Equivalence relation involving equality of dependent classes A(x) and B(y).
|- (x = y -> A = B)   &   |- R = {<.x, y>. | A = B}   =>   |- Er R
Theorem	ersym 4412	An equivalence relation is symmetric.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- (ARB -> BRA)
Theorem	ersymb 4413	An equivalence relation is symmetric.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- (ARB <-> BRA)
Theorem	ertr 4414	An equivalence relation is transitive.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- Er R   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	erref 4415	An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56.
|- Er R   =>   |- (A e. (dom R u. ran R) -> ARA)
Theorem	erdmrn 4416	The range and domain of an equivalence relation are equal.
|- Er R   =>   |- dom R = ran R
Theorem	eceq1 4417	Equality theorem for equivalence class.
|- (A = B -> [C]A = [C]B)
Theorem	eceq2 4418	Equality theorem for equivalence class.
|- (A = B -> [A]C = [B]C)
Theorem	elec 4419	Membership in an equivalence class. Theorem 72 of [Suppes] p. 82.
|- A e. V   &   |- B e. V   =>   |- (A e. [B]R <-> BRA)
Theorem	ecdmn0 4420	A representative of a nonempty equivalence class belongs to the domain of the equivalence relation.
|- A e. V   =>   |- (A e. dom R <-> [A]R =/= (/))
Theorem	erthi 4421	Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- (ARB -> [A]R = [B]R)
Theorem	erth 4422	Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82.
|- B e. V   &   |- Er R   =>   |- (A e. (dom R u. ran R) -> ([A]R = [B]R <-> ARB))
Theorem	erthdm 4423	Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership in the domain instead of just the field.
|- B e. V   &   |- Er R   =>   |- (A e. dom R -> ([A]R = [B]R <-> ARB))
Theorem	erthdmr 4424	Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- (B e. dom R -> ([A]R = [B]R <-> ARB))
Theorem	ereldm 4425	Equality of equivalence classes implies equivalence of domain membership.
|- A e. V   &   |- B e. V   &   |- Er R   &   |- dom R = D   =>   |- ([A]R = [B]R -> (A e. D <-> B e. D))
Theorem	erdisj 4426	Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- ([A]R = [B]R \/ ([A]R i^i [B]R) = (/))
Theorem	ecidsn 4427	An equivalence class modulo the identity relation is a singleton.
|- [A]I = {A}
Theorem	qseq1 4428	Equality theorem for quotient set.
|- (A = B -> (A/.C) = (B/.C))
Theorem	qseq2 4429	Equality theorem for quotient set.
|- (A = B -> (C/.A) = (C/.B))
Theorem	elqs 4430	Membership in a quotient set.
|- B e. V   =>   |- (B e. (A/.R) <-> E.x e. A B = [x]R)
Theorem	elqsi 4431	Membership in a quotient set.
|- (B e. (A/.R) -> E.x e. A B = [x]R)
Theorem	ecelqsi 4432	Membership of an equivalence class in a quotient set.
|- R e. V   =>   |- (B e. A -> [B]R e. (A/.R))
Theorem	ecopqsi 4433	"Closure" law for equivalence class of ordered pairs.
|- R e. V   &   |- S = ((A X. A)/.R)   =>   |- ((B e. A /\ C e. A) -> [<.B, C>.]R e. S)
Theorem	qsexg 4434	A quotient set exists. (Contributed by FL, 19-May-2007.)
|- (A e. V -> (A/.R) e. V)
Theorem	qsex 4435	A quotient set exists.
|- A e. V   =>   |- (A/.R) e. V
Theorem	uniqs 4436	The union of a quotient set.
|- R e. V   =>   |- U.(A/.R) = (R"A)
Theorem	snec 4437	The singleton of an equivalence class.
|- A e. V   =>   |- {[A]R} = ({A}/.R)
Theorem	ecqs 4438	Equivalence class in terms of quotient set.
|- R e. V   =>   |- [A]R = U.({A}/.R)
Theorem	0nelqs 4439	A quotient set doesn't contain the empty set.
|- dom R = A   =>   |- -. (/) e. (A/.R)
Theorem	ecelqsdm 4440	Membership of an equivalence class in a quotient set.
|- B e. V   &   |- dom R = A   =>   |- ([B]R e. (A/.R) -> B e. A)
Theorem	ecid 4441	A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.)
|- A e. V   =>   |- [A]`'E = A
Theorem	qsid 4442	A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.)
|- (A/.`'E) = A
Theorem	ectocl 4443	Implicit substitution of class for equivalence class.
|- S = (B/.R)   &   |- ([x]R = A -> (ph <-> ps))   &   |- (x e. B -> ph)   =>   |- (A e. S -> ps)
Theorem	ecoptocl 4444	Implicit substitution of class for equivalence class of ordered pair.
|- S = ((B X. C)/.R)   &   |- ([<.x, y>.]R = A -> (ph <-> ps))   &   |- ((x e. B /\ y e. C) -> ph)   =>   |- (A e. S -> ps)
Theorem	2ecoptocl 4445	Implicit substitution of classes for equivalence classes of ordered pairs.
|- S = ((C X. D)/.R)   &   |- ([<.x, y>.]R = A -> (ph <-> ps))   &   |- ([<.z, w>.]R = B -> (ps <-> ch))   &   |- (((x e. C /\ y e. D) /\ (z e. C /\ w e. D)) -> ph)   =>   |- ((A e. S /\ B e. S) -> ch)
Theorem	3ecoptocl 4446	Implicit substitution of classes for equivalence classes of ordered pairs.
|- S = ((D X. D)/.R)   &   |- ([<.x, y>.]R = A -> (ph <-> ps))   &   |- ([<.z, w>.]R = B -> (ps <-> ch))   &   |- ([<.v, u>.]R = C -> (ch <-> th))   &   |- (((x e. D /\ y e. D) /\ (z e. D /\ w e. D) /\ (v e. D /\ u e. D)) -> ph)   =>   |- ((A e. S /\ B e. S /\ C e. S) -> th)
Theorem	brecop 4447	Binary relation on a quotient set. Lemma for real number construction.
|- S e. V   &   |- Er S   &   |- dom S = (G X. G)   &   |- H = ((G X. G)/.S)   &   |- R = {<.x, y>. | ((x e. H /\ y e. H) /\ E.zE.wE.vE.u((x = [<.z, w>.]S /\ y = [<.v, u>.]S) /\ ph))}   &   |- ((((z e. G /\ w e. G) /\ (A e. G /\ B e. G)) /\ ((v e. G /\ u e. G) /\ (C e. G /\ D e. G))) -> (([<.z, w>.]S = [<.A, B>.]S /\ [<.v, u>.]S = [<.C, D>.]S) -> (ph <-> ps)))   =>   |- (((A e. G /\ B e. G) /\ (C e. G /\ D e. G)) -> ([<.A, B>.]SR[<.C, D>.]S <-> ps))
Theorem	brecop2 4448	Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis.
|- S e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   &   |- dom S = (G X. G)   &   |- H = ((G X. G)/.S)   &   |- R (_ (H X. H)   &   |- Q (_ (G X. G)   &   |- -. (/) e. G   &   |- dom F = (G X. G)   &   |- (((A e. G /\ B e. G) /\ (C e. G /\ D e. G)) -> ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC)))   =>   |- ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC))
Theorem	ecopopreq 4449	This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation R (specified by the hypothesis) in terms of its operation F.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   =>   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> (AFD) = (BFC)))
Theorem	ecopoprdm 4450	Assuming the operation F is commutative, compute the domain the relation R specified by the first hypothesis.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   &   |- (xFy) = (yFx)   =>   |- dom R = (S X. S)
Theorem	ecopoprsym 4451	Assuming the operation F is commutative, show that the relation R, specified by the first hypothesis, is symmetric.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   &   |- (xFy) = (yFx)   &   |- B e. V   =>   |- (ARB -> BRA)
Theorem	ecopoprtrn 4452	Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation R, specified by the first hypothesis, is transitive.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   &   |- (xFy) = (yFx)   &   |- ((x e. S /\ y e. S) -> (xFy) e. S)   &   |- ((xFy)Fz) = (xF(yFz))   &   |- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))   &   |- B e. V   &   |- C e. V   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	ecopoprer 4453	Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation R, specified by the first hypothesis, is an equivalence relation.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   &   |- (xFy) = (yFx)   &   |- ((x e. S /\ y e. S) -> (xFy) e. S)   &   |- ((xFy)Fz) = (xF(yFz))   &   |- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))   =>   |- Er R
Theorem	eceqopreq 4454	Equality of equivalence relation in terms of an operation.
|- B e. V   &   |- C e. V   &   |- D e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- dom F = (S X. S)   &   |- -. (/) e. S   &   |- ((x e. S /\ y e. S) -> (xFy) e. S)   &   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> (AFD) = (BFC)))   =>   |- ((A e. S /\ C e. S) -> ([<.A, B>.]R = [<.C, D>.]R <-> (AFD) = (BFC)))
Theorem	th3qlem1 4455	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption.
Theorem	th3qlem2 4456	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption.
Theorem	th3qcor 4457	Corollary of Theorem 3Q of [Enderton] p. 60.
|- R e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- ((((w e. S /\ v e. S) /\ (u e. S /\ t e. S)) /\ ((s e. S /\ f e. S) /\ (g e. S /\ h e. S))) -> ((<.w, v>.R<.u, t>. /\ <.s, f>.R<.g, h>.) -> (<.w, v>.F<.s, f>.)R(<.u, t>.F<.g, h>.)))   &   |- G = {<.<.x, y>., z>. | ((x e. ((S X. S)/.R) /\ y e. ((S X. S)/.R)) /\ E.wE.vE.uE.t((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R))}   =>   |- Fun G
Theorem	th3q 4458	Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs.
|- R e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- ((((w e. S /\ v e. S) /\ (u e. S /\ t e. S)) /\ ((s e. S /\ f e. S) /\ (g e. S /\ h e. S))) -> ((<.w, v>.R<.u, t>. /\ <.s, f>.R<.g, h>.) -> (<.w, v>.F<.s, f>.)R(<.u, t>.F<.g, h>.)))   &   |- G = {<.<.x, y>., z>. | ((x e. ((S X. S)/.R) /\ y e. ((S X. S)/.R)) /\ E.wE.vE.uE.t((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R))}   =>   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> ([<.A, B>.]RG[<.C, D>.]R) = [(<.A, B>.F<.C, D>.)]R)
Theorem	oprec 4459	Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. (See set.mm for additional comments for the hypotheses.)
|- H e. V   &   |- K e. V   &   |- L e. V   &   |- R e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ ph))}   &   |- (((z = a /\ w = b) /\ (v = c /\ u = d)) -> (ph <-> ps))   &   |- (((z = g /\ w = h) /\ (v = t /\ u = s)) -> (ph <-> ch))   &   |- G = {<.<.x, y>., z>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = J))}   &   |- (((w = a /\ v = b) /\ (u = g /\ f = h)) -> J = K)   &   |- (((w = c /\ v = d) /\ (u = t /\ f = s)) -> J = L)   &   |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> J = H)   &   |- F = {<.<.x, y>., z>. | ((x e. Q /\ y e. Q) /\ E.aE.bE.cE.d((x = [<.a, b>.]R /\ y = [<.c, d>.]R) /\ z = [(<.a, b>.G<.c, d>.)]R))}   &   |- Q = ((S X. S)/.R)   &   |- ((((a e. S /\ b e. S) /\ (c e. S /\ d e. S)) /\ ((g e. S /\ h e. S) /\ (t e. S /\ s e. S))) -> ((ps /\ ch) -> KRL))   =>   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> ([<.A, B>.]RF[<.C, D>.]R) = [H]R)
Theorem	ecoprcom 4460	Lemma used to transfer a commutative law via an equivalence relation.
|- C = ((S X. S)/.R)   &   |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> ([<.x, y>.]RF[<.z, w>.]R) = [<.D, G>.]R)   &   |- (((z e. S /\ w e. S) /\ (x e. S /\ y e. S)) -> ([<.z, w>.]RF[<.x, y>.]R) = [<.H, J>.]R)   &   |- D = H   &   |- G = J   =>   |- ((A e. C /\ B e. C) -> (AFB) = (BFA))
Theorem	ecoprass 4461	Lemma used to transfer an associative law via an equivalence relation.
|- D = ((S X. S)/.R)   &   |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> ([<.x, y>.]RF[<.z, w>.]R) = [<.G, H>.]R)   &   |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.z, w>.]RF[<.v, u>.]R) = [<.N, Q>.]R)   &   |- (((G e. S /\ H e. S) /\ (v e. S /\ u e. S)) -> ([<.G, H>.]RF[<.v, u>.]R) = [<.J, K>.]R)   &   |- (((x e. S /\ y e. S) /\ (N e. S /\ Q e. S)) -> ([<.x, y>.]RF[<.N, Q>.]R) = [<.L, M>.]R)   &   |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> (G e. S /\ H e. S))   &   |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> (N e. S /\ Q e. S))   &   |- J = L   &   |- K = M   =>   |- ((A e. D /\ B e. D /\ C e. D) -> ((AFB)FC) = (AF(BFC)))
Theorem	ecoprdi 4462	Lemma used to transfer a distributive law via an equivalence relation.
|- D = ((S X. S)/.R)   &   |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.z, w>.]RF[<.v, u>.]R) = [<.M, N>.]R)   &   |- (((x e. S /\ y e. S) /\ (M e. S /\ N e. S)) -> ([<.x, y>.]RG[<.M, N>.]R) = [<.H, J>.]R)   &   |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> ([<.x, y>.]RG[<.z, w>.]R) = [<.W, X>.]R)   &   |- (((x e. S /\ y e. S) /\ (v e. S /\ u e. S)) -> ([<.x, y>.]RG[<.v, u>.]R) = [<.Y, Z>.]R)   &   |- (((W e. S /\ X e. S) /\ (Y e. S /\ Z e. S)) -> ([<.W, X>.]RF[<.Y, Z>.]R) = [<.K, L>.]R)   &   |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> (M e. S /\ N e. S))   &   |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> (W e. S /\ X e. S))   &   |- (((x e. S /\ y e. S) /\ (v e. S /\ u e. S)) -> (Y e. S /\ Z e. S))   &   |- H = K   &   |- J = L   =>   |- ((A e. D /\ B e. D /\ C e. D) -> (AG(BFC)) = ((AGB)F(AGC)))
The mapping operation
Syntax	cm 4463	Extend the definition of a class to include the mapping operation. (Read for A ^m B, "the set of all functions that map from B to A.)
class ^m
Syntax	cpm 4464	Extend the definition of a class to include the partial mapping operation. (Read for A ^m B, "the set of all partial functions that map from B to A.)
class ^pm
Definition	df-map 4465	Define the mapping operation or set exponentiation. The set of all functions that map from B to A is written (A ^m B) (see mapval 4473). Many authors write A followed by B as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g. Definition 10.42 of [TakeutiZaring] p. 95). Other authors show B as a prefixed superscript, which is read "A pre B" (e.g. definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map(B, A) for our (A ^m B). The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation.
|- ^m = {<.<.x, y>., z>. | z = {f | f:y-->x}}
Definition	df-pm 4466	Define the partial mapping operation. A partial function from B to A is a function from a subset of B to A. The set of all partial functions from B to A is written (A ^pm B) (see pmvalg 4472). A notation for this operation apparently does not appear in the literature. We use ^pm to distinguish it from the less general set exponentiation operation ^m (df-map 4465) . See mapsspm 4480 for its relationship to set exponentiation.
|- ^pm = {<.<.x, y>., z>. | z = {f | (Fun f /\ f (_ (y X. x))}}
Theorem	mapprc 4467	When A is a proper class, the class of all functions mapping A to B is empty. Exercise 4.41 of [Mendelson] p. 255.
|- (-. A e. V -> {f | f:A-->B} = (/))
Theorem	pmex 4468	The class of all partial functions from one set to another is a set.
|- ((A e. C /\ B e. D) -> {f | (Fun f /\ f (_ (A X. B))} e. V)
Theorem	mapex 4469	The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.)
|- ((A e. C /\ B e. D) -> {f | f:A-->B} e. V)
Theorem	fnmap 4470	Set exponentiation has a universal domain.
|- ^m Fn (V X. V)
Theorem	mapvalg 4471	The value of set exponentiation. (A ^m B) is the set of all functions that map from B to A. Definition 10.24 of [Kunen] p. 24.
|- ((A e. C /\ B e. D) -> (A ^m B) = {f | f:B-->A})
Theorem	pmvalg 4472	The value of the partial mapping operation. (A ^pm B) is the set of all partial functions that map from B to A.
|- ((A e. C /\ B e. D) -> (A ^pm B) = {f | (Fun f /\ f (_ (B X. A))})
Theorem	mapval 4473	The value of set exponentiation (inference version). (A ^m B) is the set of all functions that map from B to A. Definition 10.24 of [Kunen] p. 24.
|- A e. V   &   |- B e. V   =>   |- (A ^m B) = {f | f:B-->A}
Theorem	elmapg 4474	Membership relation for set exponentiation.
|- ((A e. R /\ B e. S) -> (C e. (A ^m B) <-> C:B-->A))
Theorem	elmap 4475	Membership relation for set exponentiation.
|- A e. V   &   |- B e. V   =>   |- (F e. (A ^m B) <-> F:B-->A)
Theorem	mapval2 4476	Alternate expression for the value of set exponentiation.
|- A e. V   &   |- B e. V   =>   |- (A ^m B) = (P~(B X. A) i^i {f | f Fn B})
Theorem	elpm 4477	The predicate "is a partial function."
|- A e. V   &   |- B e. V   =>   |- (F e. (A ^pm B) <-> (Fun F /\ F (_ (B X. A)))
Theorem	elpm2 4478	The predicate "is a partial function."
|- A e. V   &   |- B e. V   =>   |- (F e. (A ^pm B) <-> (F:dom F-->A /\ dom F (_ B))
Theorem	fpm 4479	A total function is a partial function.
|- A e. V   &   |- B e. V   =>   |- (F:A-->B -> F e. (B ^pm A))
Theorem	mapsspm 4480	Set exponentiation is a subset of partial maps.
|- A e. V   &   |- B e. V   =>   |- (A ^m B) (_ (A ^pm B)
Theorem	fvopabf4 4481	Special case of fvopab4 3891 for operator theorems.
|- C e. V   &   |- D e. V   &   |- R e. V   &   |- (x = A -> B = C)   &   |- F = {<.x, y>. | (x:D-->R /\ y = B)}   =>   |- (A:D-->R -> (F` A) = C)
Theorem	mapsspw 4482	Set exponentiation is a subset of the power set of the cross product of its arguments.
|- (B e. R -> (A ^m B) (_ P~(B X. A))
Theorem	map0e 4483	Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255.
|- A e. V   =>   |- (A ^m (/)) = 1o
Theorem	map0b 4484	Set exponentiation with an empty base is the empty set, provided the exponent is non-empty. Theorem 96 of [Suppes] p. 89.
|- A e. V   =>   |- (A =/= (/) -> ((/) ^m A) = (/))
Theorem	map0 4485	Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89.
|- A e. V   &   |- B e. V   =>   |- ((A ^m B) = (/) <-> (A = (/) /\ B =/= (/)))
Theorem	mapsn 4486	The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89.
|- A e. V   &   |- B e. V   =>   |- (A ^m {B}) = {f | E.y e. A f = {<.B, y>.}}
Theorem	mapss 4487	Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89.
|- B e. V   &   |- C e. V   =>   |- (A (_ B -> (A ^m C) (_ (B ^m C))
Infinite Cartesian products
Syntax	cixp 4488	Extend class notation to include infinite Cartesian products.
class X_x e. A B
Definition	df-ixp 4489	Definition of infinite Cartesian product of [Enderton] p. 54. Enderton uses a bold "X" with x e. A written underneath or as a subscript, as does Stoll p. 47. Some books use a capital pi, but we will reserve that notation for products of numbers. Usually B represents a class expression containing x free and thus can be thought of as B(x). Normally, x is not free in A, although this is not a requirement of the definition.
|- X_x e. A B = {f | (f Fn A /\ A.x e. A (f` x) e. B)}
Theorem	elixp2 4490	Membership in an infinite Cartesian product. See df-ixp 4489 for discussion of the notation.
|- (F e. X_x e. A B <-> (F e. V /\ F Fn A /\ A.x e. A (F` x) e. B))
Theorem	elixp 4491	Membership in an infinite Cartesian product.
|- F e. V   =>   |- (F e. X_x e. A B <-> (F Fn A /\ A.x e. A (F` x) e. B))
Theorem	elixpconst 4492	Membership in an infinite Cartesian product of a constant B.
|- F e. V   =>   |- (F e. X_x e. A B <-> F:A-->B)
Theorem	ixpconst 4493	Infinite Cartesian product of a constant B.
|- A e. V   &   |- B e. V   =>   |- X_x e. A B = (B ^m A)
Theorem	ixpeq1 4494	Equality theorem for infinite Cartesian product.
|- (A = B -> X_x e. A C = X_x e. B C)
Theorem	ss2ixp 4495	Subclass theorem for infinite Cartesian product.
|- (A.x e. A B (_ C -> X_x e. A B (_ X_x e. A C)
Theorem	ixpeq2 4496	Equality theorem for infinite Cartesian product.
|- (A.x e. A B = C -> X_x e. A B = X_x e. A C)
Theorem	ixpf 4497	A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54.
|- (F e. X_x e. A B -> F:A-->U_x e. A B)
Theorem	uniixp 4498	The union of an infinite Cartesian product is included in a cross product.
|- U.X_x e. A B (_ (A X. U_x e. A B)
Theorem	ixpexg 4499	The existence of an infinite Cartesian product. x is normally a free-variable parameter in B. Remark in Enderton p. 54.
|- ((A e. C /\ A.x e. A B e. D) -> X_x e. A B e. V)
Theorem	ixp0x 4500	An infinite Cartesian product with an empty index set.
|- X_x e. (/) A = {(/)}