mmtheorems35 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3401-3500 - Page 35 of 123

Type	Label	Description
Statement
Theorem	dmss 3401	Subset theorem for domain.
|- (A (_ B -> dom A (_ dom B)
Theorem	dmeq 3402	Equality theorem for domain.
|- (A = B -> dom A = dom B)
Theorem	dmeqi 3403	Equality inference for domain.
|- A = B   =>   |- dom A = dom B
Theorem	dmeqd 3404	Equality deduction for domain.
|- (ph -> A = B)   =>   |- (ph -> dom A = dom B)
Theorem	opeldm 3405	Membership of first of an ordered pair in a domain.
|- A e. V   =>   |- (<.A, B>. e. C -> A e. dom C)
Theorem	breldm 3406	Membership of first of a binary relation in a domain.
|- A e. V   =>   |- (ARB -> A e. dom R)
Theorem	breldmg 3407	Membership of first of a binary relation in a domain.
|- ((A e. C /\ ARB) -> A e. dom R)
Theorem	dmun 3408	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 3409	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 3410	The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
|- dom U. A = U_x e. A dom x
Theorem	dmopab 3411	The domain of a class of ordered pairs.
|- dom {<.x, y>. | ph} = {x | E.yph}
Theorem	dmopabss 3412	Upper bound for the domain of a restricted class of ordered pairs.
|- dom {<.x, y>. | (x e. A /\ ph)} (_ A
Theorem	dmopab3 3413	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 3414	The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.
|- dom (/) = (/)
Theorem	dmi 3415	The domain of the identity relation is the universe.
|- dom I = V
Theorem	dmv 3416	The domain of the universe is the universe.
|- dom V = V
Theorem	dm0rn0 3417	An empty domain implies an empty range.
|- (dom A = (/) <-> ran A = (/))
Theorem	reldm0 3418	A relation is empty iff its domain is empty.
|- (Rel A -> (A = (/) <-> dom A = (/)))
Theorem	dmxp 3419	The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37.
|- (B =/= (/) -> dom ( A X. B) = A)
Theorem	dmxpid 3420	The domain of a square cross product.
|- dom ( A X. A) = A
Theorem	dmxpin 3421	The domain of the intersection of two square cross products. Unlike dmin 3409, equality holds.
|- dom ((A X. A) i^i (B X. B)) = (A i^i B)
Theorem	xpid11 3422	The cross product of a class with itself is one-to-one.
|- ((A X. A) = (B X. B) <-> A = B)
Theorem	dmcnvcnv 3423	The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 3569).
|- dom `'`'A = dom A
Theorem	rncnvcnv 3424	The range of the double converse of a class.
|- ran `'`'A = ran A
Theorem	elreldm 3425	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 3426	Equality theorem for range.
|- (A = B -> ran A = ran B)
Theorem	rneqi 3427	Equality inference for range.
|- A = B   =>   |- ran A = ran B
Theorem	rneqd 3428	Equality deduction for range.
|- (ph -> A = B)   =>   |- (ph -> ran A = ran B)
Theorem	rnss 3429	Subset theorem for range.
|- (A (_ B -> ran A (_ ran B)
Theorem	brelrng 3430	The second argument of a binary relation belongs to its range.
|- ((A e. F /\ B e. G /\ ACB) -> B e. ran C)
Theorem	brelrn 3431	The second argument of a binary relation belongs to its range.
|- A e. V   &   |- B e. V   =>   |- (ACB -> B e. ran C)
Theorem	opelrn 3432	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	releldm 3433	The first argument of a binary relation belongs to its domain.
|- ((Rel R /\ ARB) -> A e. dom R)
Theorem	relelrng 3434	The second argument of a binary relation belongs to its range.
|- ((B e. C /\ Rel R /\ ARB) -> B e. ran R)
Theorem	dfrnf 3435	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 3436	Membership in a range.
|- A e. V   =>   |- (A e. ran B <-> E.x<.x, A>. e. B)
Theorem	elrn 3437	Membership in a range.
|- A e. V   =>   |- (A e. ran B <-> E.x xBA)
Theorem	hbrn 3438	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 3439	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 3440	The range of a class of ordered pairs.
|- ran {<.x, y>. | ph} = {y | E.xph}
Theorem	rnopab2 3441	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 3442	The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.
|- ran (/) = (/)
Theorem	relrn0 3443	A relation is empty iff its range is empty.
|- (Rel A -> (A = (/) <-> ran A = (/)))
Theorem	dmrnssfld 3444	The domain and range of a class are included in its double union.
|- (dom A u. ran A) (_ U.U.A
Theorem	dmexg 3445	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 3446	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	dmex 3447	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26.
|- A e. V   =>   |- dom A e. V
Theorem	rnex 3448	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. V   =>   |- ran A e. V
Theorem	iprc 3449	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 7976.
|- -. I e. V
Theorem	dmcoss 3450	Domain of a composition. Theorem 21 of [Suppes] p. 63.
|- dom ( A o. B) (_ dom B
Theorem	rncoss 3451	Range of a composition.
|- ran ( A o. B) (_ ran A
Theorem	dmcosseq 3452	Domain of a composition.
|- (ran B (_ dom A -> dom ( A o. B) = dom B)
Theorem	dmcoeq 3453	Domain of a composition.
|- (dom A = ran B -> dom ( A o. B) = dom B)
Theorem	rncoeq 3454	Range of a composition.
|- (dom A = ran B -> ran ( A o. B) = ran A)
Theorem	reseq1 3455	Equality theorem for restrictions.
|- (A = B -> (A |` C) = (B |` C))
Theorem	reseq2 3456	Equality theorem for restrictions.
|- (A = B -> (C |` A) = (C |` B))
Theorem	hbres 3457	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 3458	A restriction to the empty set is empty.
|- (A |` (/)) = (/)
Theorem	opelres 3459	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 3460	Binary relation on a restriction.
|- B e. V   =>   |- (A(C |` D)B <-> (ACB /\ A e. D))
Theorem	opelresg 3461	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 3462	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 3463	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 3464	The restriction of a restriction.
|- ((A |` B) |` C) = (A |` (B i^i C))
Theorem	resundi 3465	Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65.
|- (A |` (B u. C)) = ((A |` B) u. (A |` C))
Theorem	resundir 3466	Distributive law for restriction over union.
|- ((A u. B) |` C) = ((A |` C) u. (B |` C))
Theorem	resindi 3467	Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
|- (A |` (B i^i C)) = ((A |` B) i^i (A |` C))
Theorem	resindir 3468	Class restriction distributes over intersection.
|- ((A i^i B) |` C) = ((A |` C) i^i (B |` C))
Theorem	inres 3469	Move intersection into class restriction.
|- (A i^i (B |` C)) = ((A i^i B) |` C)
Theorem	dmres 3470	The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25.
|- dom ( A |` B) = (B i^i dom A)
Theorem	ssdmres 3471	A domain restricted to a subclass equals the subclass.
|- (A (_ dom B <-> dom ( B |` A) = A)
Theorem	dmresexg 3472	The domain of a restriction to a set exists.
|- (B e. C -> dom ( A |` B) e. V)
Theorem	resss 3473	A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25.
|- (A |` B) (_ A
Theorem	rescom 3474	Commutative law for restriction.
|- ((A |` B) |` C) = ((A |` C) |` B)
Theorem	ssres 3475	Subclass theorem for restriction.
|- (A (_ B -> (A |` C) (_ (B |` C))
Theorem	ssres2 3476	Subclass theorem for restriction.
|- (A (_ B -> (C |` A) (_ (C |` B))
Theorem	relres 3477	A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25.
|- Rel (A |` B)
Theorem	resabs1 3478	Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.
|- (B (_ C -> ((A |` C) |` B) = (A |` B))
Theorem	resabs2 3479	Absorption law for restriction.
|- (B (_ C -> ((A |` B) |` C) = (A |` B))
Theorem	residm 3480	Idempotent law for restriction.
|- ((A |` B) |` B) = (A |` B)
Theorem	resima 3481	A restriction to an image.
|- ((A |` B)"B) = (A"B)
Theorem	relssres 3482	Simplification law for restriction.
|- ((Rel A /\ dom A (_ B) -> (A |` B) = A)
Theorem	resdm 3483	A relation restricted to its domain equals itself.
|- (Rel A -> (A |` dom A) = A)
Theorem	resexg 3484	The restriction of a set is a set.
|- (A e. C -> (A |` B) e. V)
Theorem	resopab 3485	Restriction of a class abstraction of ordered pairs.
|- ({<.x, y>. | ph} |` A) = {<.x, y>. | (x e. A /\ ph)}
Theorem	resiexg 3486	The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 3685).
|- (A e. B -> (I |` A) e. V)
Theorem	iss 3487	A subclass of the identity function is the identity function restricted to its domain.
|- (A (_ I <-> A = (I |` dom A))
Theorem	resopab2 3488	Restriction of a class abstraction of ordered pairs.
|- (A (_ B -> ({<.x, y>. | (x e. B /\ ph)} |` A) = {<.x, y>. | (x e. A /\ ph)})
Theorem	dmresi 3489	The domain of a restricted identity function.
|- dom ( I |` A) = A
Theorem	resid 3490	Any relation restricted to the universe is itself.
|- (Rel A -> (A |` V) = A)
Theorem	imaeq1 3491	Equality theorem for image.
|- (A = B -> (A"C) = (B"C))
Theorem	imaeq2 3492	Equality theorem for image.
|- (A = B -> (C"A) = (C"B))
Theorem	imaeq1i 3493	Equality theorem for image.
|- A = B   =>   |- (A"C) = (B"C)
Theorem	imaeq2i 3494	Equality theorem for image.
|- A = B   =>   |- (C"A) = (C"B)
Theorem	imaeq1d 3495	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- (ph -> A = B)   =>   |- (ph -> (A"C) = (B"C))
Theorem	imaeq2d 3496	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- (ph -> A = B)   =>   |- (ph -> (C"A) = (C"B))
Theorem	dfima2 3497	Alternate definition of image. Compare definition (d) of [Enderton] p. 44.
|- (A"B) = {y | E.x e. B xAy}
Theorem	dfima3 3498	Alternate definition of image. Compare definition (d) of [Enderton] p. 44.
|- (A"B) = {y | E.x(x e. B /\ <.x, y>. e. A)}
Theorem	elimag 3499	Membership in an image. Theorem 34 of [Suppes] p. 65.
|- (A e. D -> (A e. (B"C) <-> E.x e. C xBA))
Theorem	elima 3500	Membership in an image. Theorem 34 of [Suppes] p. 65.
|- A e. V   =>   |- (A e. (B"C) <-> E.x e. C xBA)