mmtheorems34 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3301-3400 - Page 34 of 107

Type	Label	Description
Statement
Theorem	cnvuni 3301	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 3302	Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.
|- dom A = {x | E.y<.x, y>. e. A}
Theorem	dfrn2 3303	Alternate definition of range. Definition 4 of [Suppes] p. 60.
|- ran A = {y | E.x xAy}
Theorem	dfrn3 3304	Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.
|- ran A = {y | E.x<.x, y>. e. A}
Theorem	dfdm4 3305	Alternate definition of domain.
|- dom A = ran `' A
Theorem	dfdmf 3306	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 3307	Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. V   =>   |- (A e. dom B <-> E.y ABy)
Theorem	eldm2 3308	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 3309	Domain membership. Theorem 4 of [Suppes] p. 59.
|- (A e. C -> (A e. dom B <-> E.y<.A, y>. e. B))
Theorem	dmss 3310	Subset theorem for domain.
|- (A (_ B -> dom A (_ dom B)
Theorem	dmeq 3311	Equality theorem for domain.
|- (A = B -> dom A = dom B)
Theorem	dmeqi 3312	Equality inference for domain.
|- A = B   =>   |- dom A = dom B
Theorem	dmeqd 3313	Equality deduction for domain.
|- (ph -> A = B)   =>   |- (ph -> dom A = dom B)
Theorem	opeldm 3314	Membership of first of an ordered pair in a domain.
|- A e. V   =>   |- (<.A, B>. e. C -> A e. dom C)
Theorem	breldm 3315	Membership of first of a binary relation in a domain.
|- A e. V   =>   |- (ARB -> A e. dom R)
Theorem	breldmg 3316	Membership of first of a binary relation in a domain.
|- ((A e. C /\ ARB) -> A e. dom R)
Theorem	dmun 3317	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 3318	The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60.
|- dom ( A i^i B) (_ (dom A i^i dom B)
Theorem	dmuni 3319	The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
|- dom U. A = U_x e. A dom x
Theorem	dmopab 3320	The domain of a class of ordered pairs.
|- dom {<.x, y>. | ph} = {x | E.yph}
Theorem	dmopabss 3321	Upper bound for the domain of a restricted class of ordered pairs.
|- dom {<.x, y>. | (x e. A /\ ph)} (_ A
Theorem	dmopab3 3322	The domain of a restricted class of ordered pairs.
|- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)
Theorem	dm0 3323	The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.
|- dom (/) = (/)
Theorem	dmsn0 3324	The domain of the singleton of the empty set is empty.
|- dom {(/)} = (/)
Theorem	dmsnsn0 3325	The domain of the singleton of the singleton of the empty set is empty.
|- dom {{(/)}} = (/)
Theorem	dmi 3326	The domain of the identity relation is the universe.
|- dom I = V
Theorem	dmv 3327	The domain of the universe is the universe.
|- dom V = V
Theorem	dmsnop 3328	The domain of a singleton of an ordered pair is the singleton of the first member.
|- dom {<.A, B>.} = {A}
Theorem	dmsnsnsn 3329	The domain of the singleton of the singleton of a singleton.
|- dom {{{A}}} = {A}
Theorem	dm0rn0 3330	An empty domain implies an empty range.
|- (dom A = (/) <-> ran A = (/))
Theorem	reldm0 3331	A relation is empty iff its domain is empty.
|- (Rel A -> (A = (/) <-> dom A = (/)))
Theorem	dmxp 3332	The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37.
|- (B =/= (/) -> dom ( A X. B) = A)
Theorem	dmxpid 3333	The domain of a square cross product.
|- dom ( A X. A) = A
Theorem	dmxpin 3334	The domain of the intersection of two square cross products. Unlike dmin 3318, equality holds.
|- dom ((A X. A) i^i (B X. B)) = (A i^i B)
Theorem	xpid11 3335	The cross product of a class with itself is one-to-one.
|- ((A X. A) = (B X. B) <-> A = B)
Theorem	dmcnvcnv 3336	The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 3485).
|- dom `'`'A = dom A
Theorem	rncnvcnv 3337	The range of the double converse of a class.
|- ran `'`'A = ran A
Theorem	elreldm 3338	The first member of an ordered pair in a relation belongs to the domain of the relation.
|- ((Rel A /\ B e. A) -> |^||^|B e. dom A)
Theorem	rneq 3339	Equality theorem for range.
|- (A = B -> ran A = ran B)
Theorem	rneqi 3340	Equality inference for range.
|- A = B   =>   |- ran A = ran B
Theorem	rneqd 3341	Equality deduction for range.
|- (ph -> A = B)   =>   |- (ph -> ran A = ran B)
Theorem	rnss 3342	Subset theorem for range.
|- (A (_ B -> ran A (_ ran B)
Theorem	brelrng 3343	The second argument of a binary relation belongs to its range.
|- ((A e. F /\ B e. G /\ ACB) -> B e. ran C)
Theorem	brelrnOLD 3344	The second argument of a binary relation belongs to its range.
|- A e. V   &   |- B e. V   =>   |- (ACB -> B e. ran C)
Theorem	brelrn 3345	The second argument of a binary relation belongs to its range.
|- A e. V   &   |- B e. V   =>   |- (ACB -> B e. ran C)
Theorem	opelrn 3346	Membership of second member of an ordered pair in a range.
|- A e. V   &   |- B e. V   =>   |- (<.A, B>. e. C -> B e. ran C)
Theorem	dfrnf 3347	Definition of range, 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)   =>   |- ran A = {y | E.x<.x, y>. e. A}
Theorem	elrn2 3348	Membership in a range.
|- A e. V   =>   |- (A e. ran B <-> E.x<.x, A>. e. B)
Theorem	elrn 3349	Membership in a range.
|- A e. V   =>   |- (A e. ran B <-> E.x xBA)
Theorem	hbrn 3350	Bound-variable hypothesis builder for range.
|- (y e. A -> A.x y e. A)   =>   |- (y e. ran A -> A.x y e. ran A)
Theorem	hbdm 3351	Bound-variable hypothesis builder for domain.
|- (y e. A -> A.x y e. A)   =>   |- (y e. dom A -> A.x y e. dom A)
Theorem	rnopab 3352	The range of a class of ordered pairs.
|- ran {<.x, y>. | ph} = {y | E.xph}
Theorem	rnopab2 3353	The range of a function expressed as a class abstraction.
|- ran {<.x, y>. | (x e. A /\ y = B)} = {y | E.x e. A y = B}
Theorem	rn0 3354	The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.
|- ran (/) = (/)
Theorem	relrn0 3355	A relation is empty iff its range is empty.
|- (Rel A -> (A = (/) <-> ran A = (/)))
Theorem	dmrnssfld 3356	The domain and range of a class are included in its double union.
|- (dom A u. ran A) (_ U.U.A
Theorem	dmexg 3357	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26.
|- (A e. B -> dom A e. V)
Theorem	rnexg 3358	The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41.
|- (A e. B -> ran A e. V)
Theorem	inelv 3359	The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn 7727.
|- -. I e. V
Theorem	dmcoss 3360	Domain of a composition. Theorem 21 of [Suppes] p. 63.
|- dom ( A o. B) (_ dom B
Theorem	rncoss 3361	Range of a composition.
|- ran ( A o. B) (_ ran A
Theorem	dmcosseq 3362	Domain of a composition.
|- (ran B (_ dom A -> dom ( A o. B) = dom B)
Theorem	dmcoeq 3363	Domain of a composition.
|- (dom A = ran B -> dom ( A o. B) = dom B)
Theorem	rncoeq 3364	Range of a composition.
|- (dom A = ran B -> ran ( A o. B) = ran A)
Theorem	reseq1 3365	Equality theorem for restrictions.
|- (A = B -> (A |` C) = (B |` C))
Theorem	reseq2 3366	Equality theorem for restrictions.
|- (A = B -> (C |` A) = (C |` B))
Theorem	hbres 3367	Bound-variable hypothesis builder for restriction.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A |` B) -> A.x y e. (A |` B))
Theorem	res0 3368	A restriction to the empty set is empty.
|- (A |` (/)) = (/)
Theorem	opelres 3369	Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25.
|- B e. V   =>   |- (<.A, B>. e. (C |` D) <-> (<.A, B>. e. C /\ A e. D))
Theorem	brres 3370	Binary relation on a restriction.
|- B e. V   =>   |- (A(C |` D)B <-> (ACB /\ A e. D))
Theorem	opelresg 3371	Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25.
|- (B e. R -> (<.A, B>. e. (C |` D) <-> (<.A, B>. e. C /\ A e. D)))
Theorem	opres 3372	Ordered pair membership in a restriction when the first member belongs to the restricting class.
|- B e. V   =>   |- (A e. D -> (<.A, B>. e. (C |` D) <-> <.A, B>. e. C))
Theorem	resieq 3373	A restricted identity relation is equivalent to equality in its domain.
|- ((B e. A /\ C e. A) -> (B(I |` A)C <-> B = C))
Theorem	resres 3374	The restriction of a restriction.
|- ((A |` B) |` C) = (A |` (B i^i C))
Theorem	resundi 3375	Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65.
|- (A |` (B u. C)) = ((A |` B) u. (A |` C))
Theorem	resundir 3376	Distributive law for restriction over union.
|- ((A u. B) |` C) = ((A |` C) u. (B |` C))
Theorem	dmres 3377	The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25.
|- dom ( A |` B) = (B i^i dom A)
Theorem	ssdmres 3378	A domain restricted to a subclass equals the subclass.
|- (A (_ dom B <-> dom ( B |` A) = A)
Theorem	dmresexg 3379	The domain of a restriction to a set exists.
|- (B e. C -> dom ( A |` B) e. V)
Theorem	resss 3380	A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25.
|- (A |` B) (_ A
Theorem	rescom 3381	Commutative law for restriction.
|- ((A |` B) |` C) = ((A |` C) |` B)
Theorem	ssres 3382	Subclass theorem for restriction.