mmtheorems54 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5301-5400 - Page 54 of 175

Type	Label	Description
Statement
Theorem	nnaword1 5301	Weak ordering property of addition.
|- ((A e. om /\ B e. om) -> A C_ (A +o B))
Theorem	nnaword2 5302	Weak ordering property of addition.
|- ((A e. om /\ B e. om) -> A C_ (B +o A))
Theorem	nnmordi 5303	Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers.
|- ((A e. om /\ B e. om /\ C e. om) -> ((A e. B /\ (/) e. C) -> (C .o A) e. (C .o B)))
Theorem	nnmord 5304	Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers.
|- ((A e. om /\ B e. om /\ C e. om) -> ((A e. B /\ (/) e. C) <-> (C .o A) e. (C .o B)))
Theorem	nnmcan 5305	Cancellation law for multiplication of natural numbers.
|- (((A e. om /\ B e. om /\ C e. om) /\ (/) e. A) -> ((A .o B) = (A .o C) <-> B = C))
Theorem	nnaordex 5306	Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88.
|- ((A e. om /\ B e. om) -> (A e. B <-> E.x e. om ((/) e. x /\ (A +o x) = B)))
Theorem	nnawordex 5307	Equivalence for weak ordering of natural numbers.
|- ((A e. om /\ B e. om) -> (A C_ B <-> E.x e. om (A +o x) = B))
Theorem	oaabslem 5308	Lemma for oaabs 5309.
Theorem	oaabs 5309	Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59.
|- (((A e. om /\ B e. On) /\ om C_ B) -> (A +o B) = B)
Theorem	1onn 5310	One is a natural number.
|- 1o e. om
Theorem	2onn 5311	The ordinal 2 is a natural number.
|- 2o e. om
Theorem	nneob 5312	A natural number is even iff its successor is odd.
|- (A e. om -> (E.x e. om A = (2o .o x) <-> -. E.x e. om suc A = (2o .o x)))
Theorem	omsmolem 5313	Lemma for omsmo 5314.
Theorem	omsmo 5314	A strictly monotonic ordinal function on the set of natural numbers is one-to-one.
|- (((A C_ On /\ F:om-->A) /\ A.x e. om (F` x) e. (F` suc x)) -> F:om-1-1->A)
Equivalence relations and classes
Syntax	wer 5315	Extend the definition of a wff to include the equivalence predicate.
wff Er R
Syntax	cec 5316	Extend the definition of a class to include equivalence class.
class [A]R
Syntax	cqs 5317	Extend the definition of a class to include quotient set.
class (A/.R)
Definition	df-er 5318	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 5319 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 5333, ersymb 5331, and ertr 5332.
|- (Er R <-> (`'R u. (R o. R)) C_ R)
Theorem	dfer2 5319	Alternate definition of equivalence predicate.
|- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
Definition	df-ec 5320	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 5319). 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 5321.
|- [A]R = (R"{A})
Theorem	dfec2 5321	Alternate definition of R-coset of A. Definition 34 of [Suppes] p. 81.
|- A e. _V   =>   |- [A]R = {y | ARy}
Theorem	ecexg 5322	An equivalence class modulo a set is a set.
|- (R e. B -> [A]R e. _V)
Definition	df-qs 5323	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 5324	Equality theorem for equivalence predicate.
|- (R = S -> (Er R <-> Er S))
Theorem	ster 5325	A symmetric, transitive relation is an equivalence relation.
|- (xRy -> yRx)   &   |- ((xRy /\ yRz) -> xRz)   =>   |- Er R
Theorem	ider 5326	The identity relation is an equivalence relation. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- Er _I
Theorem	iderOLD 5327	The identity relation is an equivalence relation.
|- Er _I
Theorem	eqerlem 5328	Lemma for eqer 5329.
Theorem	eqer 5329	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 5330	An equivalence relation is symmetric.
|- A e. _V   &   |- B e. _V   &   |- Er R   =>   |- (ARB -> BRA)
Theorem	ersymb 5331	An equivalence relation is symmetric.
|- A e. _V   &   |- B e. _V   &   |- Er R   =>   |- (ARB <-> BRA)
Theorem	ertr 5332	An equivalence relation is transitive.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- Er R   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	erref 5333	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 5334	The range and domain of an equivalence relation are equal.
|- Er R   =>   |- dom R = ran R
Theorem	eceq1 5335	Equality theorem for equivalence class.
|- (A = B -> [C]A = [C]B)
Theorem	eceq2 5336	Equality theorem for equivalence class.
|- (A = B -> [A]C = [B]C)
Theorem	elec 5337	Membership in an equivalence class. Theorem 72 of [Suppes] p. 82.
|- A e. _V   &   |- B e. _V   =>   |- (A e. [B]R <-> BRA)
Theorem	ecdmn0 5338	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 5339	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 5340	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 5341	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 5342	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 5343	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 5344	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 5345	An equivalence class modulo the identity relation is a singleton.
|- [A] _I = {A}
Theorem	qseq1 5346	Equality theorem for quotient set.
|- (A = B -> (A/.C) = (B/.C))
Theorem	qseq2 5347	Equality theorem for quotient set.
|- (A = B -> (C/.A) = (C/.B))
Theorem	elqs 5348	Membership in a quotient set.
|- B e. _V   =>   |- (B e. (A/.R) <-> E.x e. A B = [x]R)
Theorem	elqsi 5349	Membership in a quotient set.
|- (B e. (A/.R) -> E.x e. A B = [x]R)
Theorem	ecelqsi 5350	Membership of an equivalence class in a quotient set.
|- R e. _V   =>   |- (B e. A -> [B]R e. (A/.R))
Theorem	ecopqsi 5351	"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 5352	A quotient set exists. (Contributed by FL, 19-May-2007.)
|- (A e. _V -> (A/.R) e. _V)
Theorem	qsex 5353	A quotient set exists.
|- A e. _V   =>   |- (A/.R) e. _V
Theorem	uniqs 5354	The union of a quotient set.
|- R e. _V   =>   |- U.(A/.R) = (R"A)
Theorem	snec 5355	The singleton of an equivalence class.
|- A e. _V   =>   |- {[A]R} = ({A}/.R)
Theorem	ecqs 5356	Equivalence class in terms of quotient set.
|- R e. _V   =>   |- [A]R = U.({A}/.R)
Theorem	0nelqs 5357	A quotient set doesn't contain the empty set.
|- dom R = A   =>   |- -. (/) e. (A/.R)
Theorem	ecelqsdm 5358	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 5359	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 5360	A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.)
|- (A/.`' _E ) = A
Theorem	ectocl 5361	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 5362	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 5363	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 5364	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 5365	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 5366	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 C_ (H X. H)   &   |- Q C_ (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 5367	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 5368	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 5369	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 5370	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 5371	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 5372	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 5373	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption.
Theorem	th3qlem2 5374	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 5375	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 5376	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 5377	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 5378	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 5379	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 5380	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	cmap 5381	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 5382	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 5383	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 5391). 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 5384	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 5390). 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 5383) . See mapsspm 5398 for its relationship to set exponentiation.
|- ^pm = {<.<.x, y>., z>. | z = {f | (Fun f /\ f C_ (y X. x))}}
Theorem	mapprc 5385	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 5386	The class of all partial functions from one set to another is a set.
|- ((A e. C /\ B e. D) -> {f | (Fun f /\ f C_ (A X. B))} e. _V)
Theorem	mapex 5387	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 5388	Set exponentiation has a universal domain.
|- ^m Fn (_V X. _V)
Theorem	mapvalg 5389	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 5390	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 C_ (B X. A))})
Theorem	mapval 5391	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 5392	Membership relation for set exponentiation.
|- ((A e. R /\ B e. S) -> (C e. (A ^m B) <-> C:B-->A))
Theorem	elmap 5393	Membership relation for set exponentiation.
|- A e. _V   &   |- B e. _V   =>   |- (F e. (A ^m B) <-> F:B-->A)
Theorem	mapval2 5394	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 5395	The predicate "is a partial function."
|- A e. _V   &   |- B e. _V   =>   |- (F e. (A ^pm B) <-> (Fun F /\ F C_ (B X. A)))
Theorem	elpm2 5396	The predicate "is a partial function."
|- A e. _V   &   |- B e. _V   =>   |- (F e. (A ^pm B) <-> (F:dom F-->A /\ dom F C_ B))
Theorem	fpm 5397	A total function is a partial function.
|- A e. _V   &   |- B e. _V   =>   |- (F:A-->B -> F e. (B ^pm A))
Theorem	mapsspm 5398	Set exponentiation is a subset of partial maps.
|- A e. _V   &   |- B e. _V   =>   |- (A ^m B) C_ (A ^pm B)
Theorem	fvopabf4 5399	Special case of fvopab4 4743 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 5400	Set exponentiation is a subset of the power set of the cross product of its arguments.
|- (B e. R -> (A ^m B) C_ ~P(B X. A))