mmtheorems33 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3201-3300 - Page 33 of 105

Type	Label	Description
Statement
Theorem	xp0r 3201	The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37.
|- ((/) X. A) = (/)
Theorem	0nelxp 3202	The empty set is not a member of a cross product.
|- -. (/) e. (A X. B)
Theorem	onxpdisj 3203	Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 3088.
|- (On i^i (V X. V)) = (/)
Theorem	onnev 3204	The class of ordinal numbers is not equal to the universe.
|- On =/= V
Theorem	releq 3205	Equality theorem for the relation predicate.
|- (A = B -> (Rel A <-> Rel B))
Theorem	releqi 3206	Equality inference for the relation predicate.
|- A = B   =>   |- (Rel A <-> Rel B)
Theorem	hbrel 3207	Bound-variable hypothesis builder for a relation.
|- (y e. A -> A.x y e. A)   =>   |- (Rel A -> A.xRel A)
Theorem	relss 3208	Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.
|- (A (_ B -> (Rel B -> Rel A))
Theorem	ssrel 3209	A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33.
|- (Rel A -> (A (_ B <-> A.xA.y(<.x, y>. e. A -> <.x, y>. e. B)))
Theorem	relssi 3210	Inference from subclass principle for relations.
|- Rel A   &   |- (<.x, y>. e. A -> <.x, y>. e. B)   =>   |- A (_ B
Theorem	relssdv 3211	Deduction from subclass principle for relations.
|- (ph -> Rel A)   &   |- (ph -> (<.x, y>. e. A -> <.x, y>. e. B))   =>   |- (ph -> A (_ B)
Theorem	eqrel 3212	Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33.
|- ((Rel A /\ Rel B) -> (A = B <-> A.xA.y(<.x, y>. e. A <-> <.x, y>. e. B)))
Theorem	eqrelriv 3213	Inference from extensionality principle for relations.
|- Rel A   &   |- Rel B   &   |- (<.x, y>. e. A <-> <.x, y>. e. B)   =>   |- A = B
Theorem	eqbrriv 3214	Inference from extensionality principle for relations.
|- Rel A   &   |- Rel B   &   |- (xAy <-> xBy)   =>   |- A = B
Theorem	elrel 3215	A member of a relation is an ordered pair.
|- ((Rel R /\ A e. R) -> E.xE.y A = <.x, y>.)
Theorem	relsn 3216	A singleton of an ordered pair is a relation.
|- A e. V   =>   |- Rel {<.A, B>.}
Theorem	relxp 3217	A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37.
|- Rel (A X. B)
Theorem	ssxp 3218	Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52.
|- ((A (_ B /\ C (_ D) -> (A X. C) (_ (B X. D))
Theorem	xpsspw 3219	A cross product is included in the power of the power of the union of its arguments.
|- (A X. B) (_ P~P~(A u. B)
Theorem	unixpss 3220	The double class union of a cross product is included in the union of its arguments.
|- U.U.(A X. B) (_ (A u. B)
Theorem	xpexg 3221	The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23.
|- ((A e. C /\ B e. D) -> (A X. B) e. V)
Theorem	xpex 3222	The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23.
|- A e. V   &   |- B e. V   =>   |- (A X. B) e. V
Theorem	relun 3223	The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25.
|- (Rel (A u. B) <-> (Rel A /\ Rel B))
Theorem	relin1 3224	The intersection with a relation is a relation.
|- (Rel A -> Rel (A i^i B))
Theorem	relin2 3225	The intersection with a relation is a relation.
|- (Rel B -> Rel (A i^i B))
Theorem	reldif 3226	A difference cutting down a relation is a relation.
|- (Rel A -> Rel (A \ B))
Theorem	reluni 3227	Union law for relations. Exercise 6 of [TakeutiZaring] p. 25 and its converse.
|- (Rel U.A <-> A.x e. A Rel x)
Theorem	relopab 3228	A class of ordered pairs is a relation.
|- Rel {<.x, y>. | ph}
Theorem	opabid2 3229	A relation expressed as an ordered pair abstraction.
|- (Rel A -> {<.x, y>. | <.x, y>. e. A} = A)
Theorem	inopab 3230	Intersection of two ordered pair class abstractions.
|- ({<.x, y>. | ph} i^i {<.x, y>. | ps}) = {<.x, y>. | (ph /\ ps)}
Theorem	inxp 3231	The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25.
|- ((A X. B) i^i (C X. D)) = ((A i^i C) X. (B i^i D))
Theorem	xpindi 3232	Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52.
|- (A X. (B i^i C)) = ((A X. B) i^i (A X. C))
Theorem	xpindir 3233	Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52.
|- ((A i^i B) X. C) = ((A X. C) i^i (B X. C))
Theorem	rel0 3234	The empty set is a relation.
|- Rel (/)
Theorem	reli 3235	The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235.
|- Rel I
Theorem	rele 3236	The membership relation is a relation.
|- Rel E
Theorem	issetid 3237	Two ways of expressing set existence.
|- (A e. V <-> AIV)
Theorem	coeq1 3238	Equality theorem for composition of two classes.
|- (A = B -> (A o. C) = (B o. C))
Theorem	coeq2 3239	Equality theorem for composition of two classes.
|- (A = B -> (C o. A) = (C o. B))
Theorem	coeq1i 3240	Equality inference for composition of two classes.
|- A = B   =>   |- (A o. C) = (B o. C)
Theorem	coeq2i 3241	Equality inference for composition of two classes.
|- A = B   =>   |- (C o. A) = (C o. B)
Theorem	coeq1d 3242	Equality deduction for composition of two classes.
|- (ph -> A = B)   =>   |- (ph -> (A o. C) = (B o. C))
Theorem	coeq2d 3243	Equality deduction for composition of two classes.
|- (ph -> A = B)   =>   |- (ph -> (C o. A) = (C o. B))
Theorem	hbco 3244	Bound-variable hypothesis builder for function value.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A o. B) -> A.x y e. (A o. B))
Theorem	opelco 3245	Ordered pair membership in a composition.
|- A e. V   &   |- B e. V   =>   |- (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB))
Theorem	brco 3246	Binary relation on a composition.
|- A e. V   &   |- B e. V   =>   |- (A(C o. D)B <-> E.x(ADx /\ xCB))
Theorem	opelcog 3247	Ordered pair membership in a composition.
|- ((A e. R /\ B e. S) -> (<.A, B>. e. (C o. D) <-> E.x(<.A, x>. e. D /\ <.x, B>. e. C)))
Theorem	cnvss 3248	Subset theorem for converse.
|- (A (_ B -> `'A (_ `'B)
Theorem	cnveq 3249	Equality theorem for converse.
|- (A = B -> `'A = `'B)
Theorem	elcnv 3250	Membership in a converse. Equation 5 of [Suppes] p. 62.
|- (A e. `'R <-> E.xE.y(A = <.x, y>. /\ yRx))
Theorem	elcnv2 3251	Membership in a converse. Equation 5 of [Suppes] p. 62.
|- (A e. `'R <-> E.xE.y(A = <.x, y>. /\ <.y, x>. e. R))
Theorem	hbcnv 3252	Bound-variable hypothesis builder for converse.
|- (y e. A -> A.x y e. A)   =>   |- (y e. `'A -> A.x y e. `'A)
Theorem	opelcnvg 3253	Ordered-pair membership in converse.
|- ((A e. C /\ B e. D) -> (<.A, B>. e. `'R <-> <.B, A>. e. R))
Theorem	brcnvg 3254	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
|- ((A e. C /\ B e. D) -> (A`'RB <-> BRA))
Theorem	opelcnv 3255	Ordered-pair membership in converse.
|- A e. V   &   |- B e. V   =>   |- (<.A, B>. e. `'R <-> <.B, A>. e. R)
Theorem	brcnv 3256	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
|- A e. V   &   |- B e. V   =>   |- (A`'RB <-> BRA)
Theorem	cnvco 3257	Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64.
|- `'(A o. B) = (`'B o. `'A)
Theorem	cnvuni 3258	The converse of a class union is the (indexed) union of the converses of its members.
|- `'U.A = U_x e. A `'x
Theorem	dfdm3 3259	Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.
|- dom A = {x | E.y<.x, y>. e. A}
Theorem	dfrn2 3260	Alternate definition of range. Definition 4 of [Suppes] p. 60.
|- ran A = {y | E.x xAy}
Theorem	dfrn3 3261	Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.
|- ran A = {y | E.x<.x, y>. e. A}
Theorem	dfdm4 3262	Alternate definition of domain.
|- dom A = ran `' A
Theorem	dfdmf 3263	Definition of domain, using bound-variable hypotheses instead of distinct variable conditions.
|- (z e. A -> A.x z e. A)   &   |- (z e. A -> A.y z e. A)   =>   |- dom A = {x | E.y<.x, y>. e. A}
Theorem	eldm 3264	Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. V   =>   |- (A e. dom B <-> E.y ABy)
Theorem	eldm2 3265	Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. V   =>   |- (A e. dom B <-> E.y<.A, y>. e. B)
Theorem	eldm2g 3266	Domain membership. Theorem 4 of [Suppes] p. 59.
|- (A e. C -> (A e. dom B <-> E.y<.A, y>. e. B))
Theorem	dmss 3267	Subset theorem for domain.
|- (A (_ B -> dom A (_ dom B)
Theorem	dmeq 3268	Equality theorem for domain.
|- (A = B -> dom A = dom B)
Theorem	dmeqi 3269	Equality inference for domain.
|- A = B   =>   |- dom A = dom B
Theorem	dmeqd 3270	Equality deduction for domain.
|- (ph -> A = B)   =>   |- (ph -> dom A = dom B)
Theorem	opeldm 3271	Membership of first of an ordered pair in a domain.
|- A e. V   =>   |- (<.A, B>. e. C -> A e. dom C)
Theorem	breldm 3272	Membership of first of a binary relation in a domain.
|- A e. V   =>   |- (ARB -> A e. dom R)
Theorem	breldmg 3273	Membership of first of a binary relation in a domain.
|- ((A e. C /\ ARB) -> A e. dom R)
Theorem	dmun 3274	The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65.
|- dom ( A u. B) = (dom A u. dom B)
Theorem	dmin 3275	The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60.
|- dom (