mmtheorems33 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3201-3300 - Page 33 of 107

Type	Label	Description
Statement
Theorem	opelxp1 3201	The first member of an ordered pair of classes in a cross product belongs to first cross product argument.
|- (<.A, B>. e. (C X. D) -> A e. C)
Theorem	otelxp1 3202	The first member of an ordered triple of classes in a cross product belongs to first cross product argument.
|- (<.<.A, B>., C>. e. ((R X. S) X. T) -> A e. R)
Theorem	brrelex 3203	A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.)
|- ((Rel R /\ ARB) -> A e. V)
Theorem	brrelexi 3204	The first argument of a binary relation exists. (An artifact of our ordered pair definition.)
|- Rel R   =>   |- (ARB -> A e. V)
Theorem	nprrel 3205	No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
|- Rel R   &   |- -. A e. V   =>   |- -. ARB
Theorem	fconstopab 3206	Representation of a constant function using ordered pairs.
|- (A X. {B}) = {<.x, y>. | (x e. A /\ y = B)}
Theorem	vtoclr 3207	Variable to class conversion of transitive relation.
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   =>   |- (C e. D -> ((ARB /\ BRC) -> ARC))
Theorem	vtoclrbr 3208	Variable to class conversion of transitive, reflexive relation.
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   &   |- xRx   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	vtoclibr 3209	Variable to class conversion of transitive, irreflexive relation.
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   &   |- -. xRx   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	opelxp 3210	Ordered pair membership in a cross product.
|- B e. V   =>   |- (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D))
Theorem	brxp 3211	Binary relation on a cross product.
|- B e. V   =>   |- (A(C X. D)B <-> (A e. C /\ B e. D))
Theorem	opelxpg 3212	Ordered pair membership in a cross product.
|- (B e. R -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))
Theorem	opelxpi 3213	Ordered pair membership in a cross product (implication).
|- ((A e. C /\ B e. D) -> <.A, B>. e. (C X. D))
Theorem	ralxp 3214	Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution.
|- (x = <.y, z>. -> (ph <-> ps))   =>   |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)
Theorem	rexxp 3215	Existential quantification restricted to a cross product is equivalent to a double restricted quantification.
|- (x = <.y, z>. -> (ph <-> ps))   =>   |- (E.x e. (A X. B)ph <-> E.y e. A E.z e. B ps)
Theorem	ralxpf 3216	Version of ralxp 3214 with bound-variable hypotheses.
|- (ph -> A.yph)   &   |- (ph -> A.zph)   &   |- (ps -> A.xps)   &   |- (x = <.y, z>. -> (ph <-> ps))   =>   |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)
Theorem	opthprc 3217	Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes."
|- (((A X. {(/)}) u. (B X. {{(/)}})) = ((C X. {(/)}) u. (D X. {{(/)}})) <-> (A = C /\ B = D))
Theorem	brelg 3218	Two things in a binary relation belong to the relation's domain.
|- R (_ (C X. D)   =>   |- (B e. S -> (ARB -> (A e. C /\ B e. D)))
Theorem	brel 3219	Membership in superset of binary relation.
|- B e. V   &   |- R (_ (C X. D)   =>   |- (ARB -> (A e. C /\ B e. D))
Theorem	elxp3 3220	Membership in a cross product.
|- (A e. (B X. C) <-> E.xE.y(<.x, y>. = A /\ <.x, y>. e. (B X. C)))
Theorem	xpundi 3221	Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52.
|- (A X. (B u. C)) = ((A X. B) u. (A X. C))
Theorem	xpundir 3222	Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52.
|- ((A u. B) X. C) = ((A X. C) u. (B X. C))
Theorem	xpun 3223	The cross product of two unions.
|- ((A u. B) X. (C u. D)) = (((A X. C) u. (A X. D)) u. ((B X. C) u. (B X. D)))
Theorem	elvv 3224	Membership in universal class of ordered pairs.
|- (A e. (V X. V) <-> E.xE.y A = <.x, y>.)
Theorem	elvvuni 3225	An ordered pair contains its union.
|- (A e. (V X. V) -> U.A e. A)
Theorem	xpss 3226	A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25.
|- (A X. B) (_ (V X. V)
Theorem	brinxp2 3227	Intersection of binary relation with cross product.
|- (B e. S -> (A(R i^i (C X. D))B <-> (A e. C /\ B e. D /\ ARB)))
Theorem	brinxp 3228	Intersection of binary relation with cross product.
|- ((A e. C /\ B e. D) -> (ARB <-> A(R i^i (C X. D))B))
Theorem	weinxp 3229	Intersection of well-ordering with cross product of its field.
|- (R We A <-> (R i^i (A X. A)) We A)
Theorem	opabssxp 3230	An abstraction relation is a subset of a related cross product.
|- {<.x, y>. | ((x e. A /\ y e. B) /\ ph)} (_ (A X. B)
Theorem	optocl 3231	Implicit substitution of class for ordered pair.
|- D = (B X. C)   &   |- (<.x, y>. = A -> (ph <-> ps))   &   |- ((x e. B /\ y e. C) -> ph)   =>   |- (A e. D -> ps)
Theorem	2optocl 3232	Implicit substitution of classes for ordered pairs.
|- R = (C X. D)   &   |- (<.x, y>. = A -> (ph <-> ps))   &   |- (<.z, w>. = B -> (ps <-> ch))   &   |- (((x e. C /\ y e. D) /\ (z e. C /\ w e. D)) -> ph)   =>   |- ((A e. R /\ B e. R) -> ch)
Theorem	3optocl 3233	Implicit substitution of classes for ordered pairs.
|- R = (D X. F)   &   |- (<.x, y>. = A -> (ph <-> ps))   &   |- (<.z, w>. = B -> (ps <-> ch))   &   |- (<.v, u>. = C -> (ch <-> th))   &   |- (((x e. D /\ y e. F) /\ (z e. D /\ w e. F) /\ (v e. D /\ u e. F)) -> ph)   =>   |- ((A e. R /\ B e. R /\ C e. R) -> th)
Theorem	opbrop 3234	Ordered pair membership in a relation. Special case.
|- (((z = A /\ w = B) /\ (v = C /\ u = D)) -> (ph <-> ps))   &   |- 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))}   =>   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> ps))
Theorem	xp0r 3235	The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37.
|- ((/) X. A) = (/)
Theorem	0nelxp 3236	The empty set is not a member of a cross product.
|- -. (/) e. (A X. B)
Theorem	onxpdisj 3237	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 3121.
|- (On i^i (V X. V)) = (/)
Theorem	onnev 3238	The class of ordinal numbers is not equal to the universe.
|- On =/= V
Theorem	releq 3239	Equality theorem for the relation predicate.
|- (A = B -> (Rel A <-> Rel B))
Theorem	releqi 3240	Equality inference for the relation predicate.
|- A = B   =>   |- (Rel A <-> Rel B)
Theorem	hbrel 3241	Bound-variable hypothesis builder for a relation.
|- (y e. A -> A.x y e. A)   =>   |- (Rel A -> A.xRel A)
Theorem	relss 3242	Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.
|- (A (_ B -> (Rel B -> Rel A))
Theorem	ssrel 3243	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 3244	Inference from subclass principle for relations.
|- Rel A   &   |- (<.x, y>. e. A -> <.x, y>. e. B)   =>   |- A (_ B
Theorem	relssdv 3245	Deduction from subclass principle for relations.
|- (ph -> Rel A)   &   |- (ph -> (<.x, y>. e. A -> <.x, y>. e. B))   =>   |- (ph -> A (_ B)
Theorem	eqrel 3246	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 3247	Inference from extensionality principle for relations.
|- Rel A   &   |- Rel B   &   |- (<.x, y>. e. A <-> <.x, y>. e. B)   =>   |- A = B
Theorem	eqbrriv 3248	Inference from extensionality principle for relations.
|- Rel A   &   |- Rel B   &   |- (xAy <-> xBy)   =>   |- A = B
Theorem	elrel 3249	A member of a relation is an ordered pair.
|- ((Rel R /\ A e. R) -> E.xE.y A = <.x, y>.)
Theorem	relsn 3250	A singleton of an ordered pair is a relation.
|- A e. V   =>   |- Rel {<.A, B