Metamath Proof Explorer - mmtheorems26

Jump to page: 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5787

Statement List for Metamath Proof Explorer - 2501-2600 - Page 26 of 58

Type	Label	Description
Statement
Theorem	rele 2501	The membership relation is a relation.
|- Rel E
Theorem	coeq1 2502	Equality theorem for composition of two classes.
|- (A = B -> (A o. C) = (B o. C))
Theorem	coeq2 2503	Equality theorem for composition of two classes.
|- (A = B -> (C o. A) = (C o. B))
Theorem	coeq1i 2504	Equality inference for composition of two classes.
|- A = B   =>   |- (A o. C) = (B o. C)
Theorem	coeq2i 2505	Equality inference for composition of two classes.
|- A = B   =>   |- (C o. A) = (C o. B)
Theorem	coeq1d 2506	Equality deduction for composition of two classes.
|- (ph -> A = B)   =>   |- (ph -> (A o. C) = (B o. C))
Theorem	coeq2d 2507	Equality deduction for composition of two classes.
|- (ph -> A = B)   =>   |- (ph -> (C o. A) = (C o. B))
Theorem	hbco 2508	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 2509	Ordered pair membership in a composition.
|- A e. V   &   |- B e. V   =>   |- (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB))
Theorem	brco 2510	Binary relation on a composition.
|- A e. V   &   |- B e. V   =>   |- (A(C o. D)B <-> E.x(ADx /\ xCB))
Theorem	opelcog 2511	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 2512	Subset theorem for converse.
|- (A (_ B -> `'A (_ `'B)
Theorem	cnveq 2513	Equality theorem for converse.
|- (A = B -> `'A = `'B)
Theorem	elcnv 2514	Membership in a converse. Equation (5) of [Suppes] p. 62.
|- (A e. `'R <-> E.xE.y(A = <.x, y>. /\ yRx))
Theorem	elcnv2 2515	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 2516	Bound-variable hypothesis builder for converse.
|- (y e. A -> A.x y e. A)   =>   |- (y e. `'A -> A.x y e. `'A)
Theorem	opelcnvg 2517	Ordered-pair membership in converse.
|- ((A e. C /\ B e. D) -> (<.A, B>. e. `'R <-> <.B, A>. e. R))
Theorem	opelcnv 2518	Ordered-pair membership in converse.
|- A e. V   &   |- B e. V   =>   |- (<.A, B>. e. `'R <-> <.B, A>. e. R)
Theorem	brcnv 2519	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 2520	Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64.
|- `'(A o. B) = (`'B o. `'A)
Theorem	cnvuni 2521	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 2522	Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.
|- dom A = {x | E.y<.x, y>. e. A}
Theorem	dfrn2 2523	Alternate definition of range. Definition 4 of [Suppes] p. 60.
|- ran A = {y | E.x xAy}
Theorem	dfrn3 2524	Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.
|- ran A = {y | E.x<.x, y>. e. A}
Theorem	dfdm4 2525	Alternate definition of domain.
|- dom A = ran `'A
Theorem	dfdmf 2526	Domain definition requiring only that x and y not be 'free' in A (but not necessarily absent from it).
|- (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 2527	Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. V   =>   |- (A e. dom B <-> E.y ABy)
Theorem	eldm2 2528	Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. V   =>   |- (A e. dom B <-> E.y<.A, y>. e. B)
Theorem	eldmg 2529	Domain membership. Theorem 4 of [Suppes] p. 59.
|- (A e. C -> (A e. dom B <-> E.y<.A, y>. e. B))
Theorem	dmss 2530	Subset theorem for domain.
|- (A (_ B -> dom A (_ dom B)
Theorem	dmeq 2531	Equality theorem for domain.
|- (A = B -> dom A = dom B)
Theorem	dmeqi 2532	Equality inference for domain.
|- A = B   =>   |- dom A = dom B
Theorem	dmeqd 2533	Equality deduction for domain.
|- (ph -> A = B)   =>   |- (ph -> dom A = dom B)
Theorem	opeldm 2534	Membership of first of an ordered pair in a domain.
|- A e. V   =>   |- (<.A, B>. e. C -> A e. dom C)
Theorem	breldm 2535	Membership of first of a binary relation in a domain.
|- A e. V   =>   |- (ARB -> A e. dom R)
Theorem	dmun 2536	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 2537	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 2538	The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
|- dom U.A = U.x e. A dom x
Theorem	dmopab 2539	The domain of a class of ordered pairs.
|- dom {<.x, y>. | ph} = {x | E.yph}
Theorem	dmopabss 2540	Upper bound for the domain of a restricted class of ordered pairs.
|- dom {<.x, y>. | (x e. A /\ ph)} (_ A
Theorem	dmopab2 2541	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 2542	The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.
|- dom (/) = (/)
Theorem	dmsn0 2543	The domain of the singleton of the empty set is empty.
|- dom {(/)} = (/)
Theorem	dmsnsn0 2544	The domain of the singleton of the singleton of the empty set is empty.
|- dom {{(/)}} = (/)
Theorem	dmi 2545	The domain of the identity relation is the universe.
|- dom I = V
Theorem	dmv 2546	The domain of the universe is the universe.
|- dom V = V
Theorem	dmsnop 2547	The domain of a singleton of an ordered pair is the singleton of the first member.
|- dom {<.A, B>.} = {A}
Theorem	dmsnsnsn 2548	The domain of the singleton of the singleton of a singleton.
|- dom {{{A}}} = {A}
Theorem	dm0rn0 2549	An empty domain implies an empty range.
|- (dom A = (/) <-> ran A = (/))
Theorem	reldm0 2550	A relation is empty iff its domain is empty.
|- (Rel A -> (A = (/) <-> dom A = (/)))
Theorem	dmexg 2551	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26.
|- (A e. B -> dom A e. V)
Theorem	dmxp 2552	The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37.
|- (-. B = (/) -> dom (A X. B) = A)
Theorem	dmxpid 2553	The domain of a cross product.
|- dom (A X. A) = A
Theorem	elreldm 2554	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 2555	Equality theorem for range.
|- (A = B -> ran A = ran B)
Theorem	rneqi 2556	Equality inference for range.
|- A = B   =>   |- ran A = ran B
Theorem	rneqd 2557	Equality deduction for range.
|- (ph -> A = B)   =>   |- (ph -> ran A = ran B)
Theorem	rnss 2558	Subset theorem for range.
|- (A (_ B -> ran A (_ ran B)
Theorem	brelrn 2559	The second argument of a binary relation belongs to its range.
|- A e. V   &   |- B e. V   =>   |- (ACB -> B e. ran C)
Theorem	opelrn 2560	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 2561	Range definition requiring only that x and y not be 'free' in A (but not necessarily absent from it).
|- (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	elrn 2562	Membership in a range.
|- A e. V   =>   |- (A e. ran B <-> E.x<.x, A>. e. B)
Theorem	elrn2 2563	Membership in a range.
|- A e. V   =>   |- (A e. ran B <-> E.x xBA)
Theorem	hbrn 2564	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 2565	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 2566	The range of a class of ordered pairs.
|- ran {<.x, y>. | ph} = {y | E.xph}
Theorem	rn0 2567	The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.
|- ran (/) = (/)
Theorem	relrn0 2568	A relation is empty iff its range is empty.
|- (Rel A -> (A = (/) <-> ran A = (/)))
Theorem	rnexg 2569	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	dmco 2570	Domain of a composition. Theorem 21 of [Suppes] p. 63.
|- dom (A o. B) (_ dom B
Theorem	rnco 2571	Range of a composition.
|- ran (A o. B) (_ ran A
Theorem	dmcosseq 2572	Domain of a composition.
|- (ran B (_ dom A -> dom (A o. B) = dom B)
Theorem	dmcoeq 2573	Domain of a composition.
|- (dom A = ran B -> dom (A o. B) = dom B)
Theorem	rncoeq 2574	Range of a composition.
|- (dom A = ran B -> ran (A o. B) = ran A)
Theorem	reseq1 2575	Equality theorem for restrictions.
|- (A = B -> (A |` C) = (B |` C))
Theorem	reseq2 2576	Equality theorem for restrictions.
|- (A = B -> (C |` A) = (C |` B))
Theorem	hbres 2577	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 2578	A restriction to the empty set is empty.
|- (A |` (/)) = (/)
Theorem	opelres 2579	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	opres 2580	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 2581	A restricted identity relation is equivalent to equality in its domain.
|- ((B e. A /\ C e. A) -> (B(I |` A)C <-> B = C