P. 54 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfcnv 5301	Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   =>    |- F/_ x `' A
Theorem	opelcnvg 5302	Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )
Theorem	brcnvg 5303	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
|- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
Theorem	opelcnv 5304	Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. e. `' R <-> <. B , A >. e. R )
Theorem	brcnv 5305	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
|- A e. _V   &    |- B e. _V   =>    |- ( A `' R B <-> B R A )
Theorem	csbcnv 5306	Move class substitution in and out of the converse of a function. Version of csbcnvgALT 5307 without a sethood antecedent but depending on more axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.)
|- `' [_ A / x ]_ F = [_ A / x ]_ `' F
Theorem	csbcnvgALT 5307	Move class substitution in and out of the converse of a function. Version of csbcnv 5306 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )
Theorem	cnvco 5308	Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- `' ( A o. B ) = ( `' B o. `' A )
Theorem	cnvuni 5309*	The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
|- `' U. A = U_ x e. A `' x
Theorem	dfdm3 5310*	Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
|- dom A = { x | E. y <. x , y >. e. A }
Theorem	dfrn2 5311*	Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
|- ran A = { y | E. x x A y }
Theorem	dfrn3 5312*	Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
|- ran A = { y | E. x <. x , y >. e. A }
Theorem	elrn2g 5313*	Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
|- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )
Theorem	elrng 5314*	Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
|- ( A e. V -> ( A e. ran B <-> E. x x B A ) )
Theorem	ssrelrn 5315*	If a relation is a subset of a cartesian product, then for each element of the range of the relation there is an element of the first set of the cartesian product which is related to the element of the range by the relation. (Contributed by AV, 24-Oct-2020.)
|- ( ( R C_ ( A X. B ) /\ Y e. ran R ) -> E. a e. A a R Y )
Theorem	dfdm4 5316	Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
|- dom A = ran `' A
Theorem	dfdmf 5317*	Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   =>    |- dom A = { x | E. y x A y }
Theorem	csbdm 5318	Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)
|- [_ A / x ]_ dom B = dom [_ A / x ]_ B
Theorem	eldmg 5319*	Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
|- ( A e. V -> ( A e. dom B <-> E. y A B y ) )
Theorem	eldm2g 5320*	Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- ( A e. V -> ( A e. dom B <-> E. y <. A , y >. e. B ) )
Theorem	eldm 5321*	Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
|- A e. _V   =>    |- ( A e. dom B <-> E. y A B y )
Theorem	eldm2 5322*	Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
|- A e. _V   =>    |- ( A e. dom B <-> E. y <. A , y >. e. B )
Theorem	dmss 5323	Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
|- ( A C_ B -> dom A C_ dom B )
Theorem	dmeq 5324	Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
|- ( A = B -> dom A = dom B )
Theorem	dmeqi 5325	Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
|- A = B   =>    |- dom A = dom B
Theorem	dmeqd 5326	Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> dom A = dom B )
Theorem	opeldmd 5327	Membership of first of an ordered pair in a domain. Deduction version of opeldm 5328. (Contributed by AV, 11-Mar-2021.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> ( <. A , B >. e. C -> A e. dom C ) )
Theorem	opeldm 5328	Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. e. C -> A e. dom C )
Theorem	breldm 5329	Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
|- A e. _V   &    |- B e. _V   =>    |- ( A R B -> A e. dom R )
Theorem	breldmg 5330	Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
|- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R )
Theorem	dmun 5331	The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- dom ( A u. B ) = ( dom A u. dom B )
Theorem	dmin 5332	The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
|- dom ( A i^i B ) C_ ( dom A i^i dom B )
Theorem	dmiun 5333	The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
|- dom U_ x e. A B = U_ x e. A dom B
Theorem	dmuni 5334*	The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
|- dom U. A = U_ x e. A dom x
Theorem	dmopab 5335*	The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
|- dom { <. x , y >. | ph } = { x | E. y ph }
Theorem	dmopabss 5336*	Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
|- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A
Theorem	dmopab3 5337*	The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
|- ( A. x e. A E. y ph <-> dom { <. x , y >. | ( x e. A /\ ph ) } = A )
Theorem	opabssxpd 5338*	An ordered-pair class abstraction is a subset of an Cartesian product. Formerly part of proof for opabex2 7227. (Contributed by AV, 26-Nov-2021.)
|- ( ( ph /\ ps ) -> x e. A )   &    |- ( ( ph /\ ps ) -> y e. B )   =>    |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )
Theorem	dm0 5339	The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- dom (/) = (/)
Theorem	dmi 5340	The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- dom _I = _V
Theorem	dmv 5341	The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
|- dom _V = _V
Theorem	dm0rn0 5342	An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.)
|- ( dom A = (/) <-> ran A = (/) )
Theorem	reldm0 5343	A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
|- ( Rel A -> ( A = (/) <-> dom A = (/) ) )
Theorem	dmxp 5344	The domain of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( B =/= (/) -> dom ( A X. B ) = A )
Theorem	dmxpid 5345	The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.)
|- dom ( A X. A ) = A
Theorem	dmxpin 5346	The domain of the intersection of two square Cartesian products. Unlike dmin 5332, equality holds. (Contributed by NM, 29-Jan-2008.)
|- dom ( ( A X. A ) i^i ( B X. B ) ) = ( A i^i B )
Theorem	xpid11 5347	The Cartesian product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( A X. A ) = ( B X. B ) <-> A = B )
Theorem	dmcnvcnv 5348	The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5583). (Contributed by NM, 8-Apr-2007.)
|- dom `' `' A = dom A
Theorem	rncnvcnv 5349	The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
|- ran `' `' A = ran A
Theorem	elreldm 5350	The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
|- ( ( Rel A /\ B e. A ) -> |^| |^| B e. dom A )
Theorem	rneq 5351	Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
|- ( A = B -> ran A = ran B )
Theorem	rneqi 5352	Equality inference for range. (Contributed by NM, 4-Mar-2004.)
|- A = B   =>    |- ran A = ran B
Theorem	rneqd 5353	Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> ran A = ran B )
Theorem	rnss 5354	Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
|- ( A C_ B -> ran A C_ ran B )
Theorem	brelrng 5355	The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
|- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )
Theorem	brelrn 5356	The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
|- A e. _V   &    |- B e. _V   =>    |- ( A C B -> B e. ran C )
Theorem	opelrn 5357	Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. e. C -> B e. ran C )
Theorem	releldm 5358	The first argument of a binary relation belongs to its domain. Note that A R B does not imply Rel R: see for example nrelv 5244 and brv 4941. (Contributed by NM, 2-Jul-2008.)
|- ( ( Rel R /\ A R B ) -> A e. dom R )
Theorem	relelrn 5359	The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
|- ( ( Rel R /\ A R B ) -> B e. ran R )
Theorem	releldmb 5360*	Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
|- ( Rel R -> ( A e. dom R <-> E. x A R x ) )
Theorem	relelrnb 5361*	Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
|- ( Rel R -> ( A e. ran R <-> E. x x R A ) )
Theorem	releldmi 5362	The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
|- Rel R   =>    |- ( A R B -> A e. dom R )
Theorem	relelrni 5363	The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
|- Rel R   =>    |- ( A R B -> B e. ran R )
Theorem	dfrnf 5364*	Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   =>    |- ran A = { y | E. x x A y }
Theorem	elrn2 5365*	Membership in a range. (Contributed by NM, 10-Jul-1994.)
|- A e. _V   =>    |- ( A e. ran B <-> E. x <. x , A >. e. B )
Theorem	elrn 5366*	Membership in a range. (Contributed by NM, 2-Apr-2004.)
|- A e. _V   =>    |- ( A e. ran B <-> E. x x B A )
Theorem	nfdm 5367	Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   =>    |- F/_ x dom A
Theorem	nfrn 5368	Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   =>    |- F/_ x ran A
Theorem	dmiin 5369	Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
|- dom |^|_ x e. A B C_ |^|_ x e. A dom B
Theorem	rnopab 5370*	The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
|- ran { <. x , y >. | ph } = { y | E. x ph }
Theorem	rnmpt 5371*	The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. A |-> B )   =>    |- ran F = { y | E. x e. A y = B }
Theorem	elrnmpt 5372*	The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
|- F = ( x e. A |-> B )   =>    |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
Theorem	elrnmpt1s 5373*	Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
|- F = ( x e. A |-> B )   &    |- ( x = D -> B = C )   =>    |- ( ( D e. A /\ C e. V ) -> C e. ran F )
Theorem	elrnmpt1 5374	Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. A |-> B )   =>    |- ( ( x e. A /\ B e. V ) -> B e. ran F )
Theorem	elrnmptg 5375*	Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. A |-> B )   =>    |- ( A. x e. A B e. V -> ( C e. ran F <-> E. x e. A C = B ) )
Theorem	elrnmpti 5376*	Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. A |-> B )   &    |- B e. _V   =>    |- ( C e. ran F <-> E. x e. A C = B )
Theorem	rn0 5377	The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.)
|- ran (/) = (/)
Theorem	dfiun3g 5378	Alternate definition of indexed union when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( A. x e. A B e. C -> U_ x e. A B = U. ran ( x e. A |-> B ) )
Theorem	dfiin3g 5379	Alternate definition of indexed intersection when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( A. x e. A B e. C -> |^|_ x e. A B = |^| ran ( x e. A |-> B ) )
Theorem	dfiun3 5380	Alternate definition of indexed union when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- B e. _V   =>    |- U_ x e. A B = U. ran ( x e. A |-> B )
Theorem	dfiin3 5381	Alternate definition of indexed intersection when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- B e. _V   =>    |- |^|_ x e. A B = |^| ran ( x e. A |-> B )
Theorem	riinint 5382*	Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( ( X e. V /\ A. k e. I S C_ X ) -> ( X i^i |^|_ k e. I S ) = |^| ( { X } u. ran ( k e. I |-> S ) ) )
Theorem	relrn0 5383	A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
|- ( Rel A -> ( A = (/) <-> ran A = (/) ) )
Theorem	dmrnssfld 5384	The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
|- ( dom A u. ran A ) C_ U. U. A
Theorem	dmcoss 5385	Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- dom ( A o. B ) C_ dom B
Theorem	rncoss 5386	Range of a composition. (Contributed by NM, 19-Mar-1998.)
|- ran ( A o. B ) C_ ran A
Theorem	dmcosseq 5387	Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ran B C_ dom A -> dom ( A o. B ) = dom B )
Theorem	dmcoeq 5388	Domain of a composition. (Contributed by NM, 19-Mar-1998.)
|- ( dom A = ran B -> dom ( A o. B ) = dom B )
Theorem	rncoeq 5389	Range of a composition. (Contributed by NM, 19-Mar-1998.)
|- ( dom A = ran B -> ran ( A o. B ) = ran A )
Theorem	reseq1 5390	Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
|- ( A = B -> ( A |` C ) = ( B |` C ) )
Theorem	reseq2 5391	Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
|- ( A = B -> ( C |` A ) = ( C |` B ) )
Theorem	reseq1i 5392	Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
|- A = B   =>    |- ( A |` C ) = ( B |` C )
Theorem	reseq2i 5393	Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = B   =>    |- ( C |` A ) = ( C |` B )
Theorem	reseq12i 5394	Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
|- A = B   &    |- C = D   =>    |- ( A |` C ) = ( B |` D )
Theorem	reseq1d 5395	Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A |` C ) = ( B |` C ) )
Theorem	reseq2d 5396	Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C |` A ) = ( C |` B ) )
Theorem	reseq12d 5397	Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A |` C ) = ( B |` D ) )
Theorem	nfres 5398	Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A |` B )
Theorem	csbres 5399	Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)
|- [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C )
Theorem	res0 5400	A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
|- ( A |` (/) ) = (/)