P. 50 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4901-5000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ralxpf 4901*	Version of ralxp 4898 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/ y ph   &    |- F/ z ph   &    |- F/ x ps   &    |- ( x = <. y , z >. -> ( ph <-> ps ) )   =>    |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps )
Theorem	rexxpf 4902*	Version of rexxp 4899 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/ y ph   &    |- F/ z ph   &    |- F/ x ps   &    |- ( x = <. y , z >. -> ( ph <-> ps ) )   =>    |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps )
Theorem	iunxpf 4903*	Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
|- F/_ y C   &    |- F/_ z C   &    |- F/_ x D   &    |- ( x = <. y , z >. -> C = D )   =>    |- U_ x e. ( A X. B ) C = U_ y e. A U_ z e. B D
Theorem	opabbi2dv 4904*	Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2353. (Contributed by NM, 24-Feb-2014.)
|- Rel A   &    |- ( ph -> ( <. x , y >. e. A <-> ps ) )   =>    |- ( ph -> A = { <. x , y >. | ps } )
Theorem	relop 4905*	A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
|- A e. _V   &    |- B e. _V   =>    |- ( Rel <. A , B >. <-> E. x E. y ( A = { x } /\ B = { x , y } ) )
Theorem	ideqg 4906	For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( B e. V -> ( A _I B <-> A = B ) )
Theorem	ideq 4907	For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
|- B e. _V   =>    |- ( A _I B <-> A = B )
Theorem	ididg 4908	A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A e. V -> A _I A )
Theorem	issetid 4909	Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A e. _V <-> A _I A )
Theorem	coss1 4910	Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
|- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )
Theorem	coss2 4911	Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
|- ( A C_ B -> ( C o. A ) C_ ( C o. B ) )
Theorem	coeq1 4912	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
|- ( A = B -> ( A o. C ) = ( B o. C ) )
Theorem	coeq2 4913	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
|- ( A = B -> ( C o. A ) = ( C o. B ) )
Theorem	coeq1i 4914	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
|- A = B   =>    |- ( A o. C ) = ( B o. C )
Theorem	coeq2i 4915	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
|- A = B   =>    |- ( C o. A ) = ( C o. B )
Theorem	coeq1d 4916	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A o. C ) = ( B o. C ) )
Theorem	coeq2d 4917	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C o. A ) = ( C o. B ) )
Theorem	coeq12i 4918	Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
|- A = B   &    |- C = D   =>    |- ( A o. C ) = ( B o. D )
Theorem	coeq12d 4919	Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A o. C ) = ( B o. D ) )
Theorem	nfco 4920	Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A o. B )
Theorem	elco 4921*	Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.)
|- ( A e. ( R o. S ) <-> E. x E. y E. z ( A = <. x , z >. /\ ( x S y /\ y R z ) ) )
Theorem	brcog 4922*	Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
|- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
Theorem	opelco2g 4923*	Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. ( C o. D ) <-> E. x ( <. A , x >. e. D /\ <. x , B >. e. C ) ) )
Theorem	brcogw 4924	Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
|- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )
Theorem	eqbrrdva 4925*	Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
|- ( ph -> A C_ ( C X. D ) )   &    |- ( ph -> B C_ ( C X. D ) )   &    |- ( ( ph /\ x e. C /\ y e. D ) -> ( x A y <-> x B y ) )   =>    |- ( ph -> A = B )
Theorem	brco 4926*	Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
|- A e. _V   &    |- B e. _V   =>    |- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) )
Theorem	opelco 4927*	Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. e. ( C o. D ) <-> E. x ( A D x /\ x C B ) )
Theorem	cnvss 4928	Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
|- ( A C_ B -> `' A C_ `' B )
Theorem	cnveq 4929	Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
|- ( A = B -> `' A = `' B )
Theorem	cnveqi 4930	Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
|- A = B   =>    |- `' A = `' B
Theorem	cnveqd 4931	Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> `' A = `' B )
Theorem	elcnv 4932*	Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
|- ( A e. `' R <-> E. x E. y ( A = <. x , y >. /\ y R x ) )
Theorem	elcnv2 4933*	Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
|- ( A e. `' R <-> E. x E. y ( A = <. x , y >. /\ <. y , x >. e. R ) )
Theorem	nfcnv 4934	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 4935	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 4936	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 4937	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 4938	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	csbcnvg 4939	Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
|- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )
Theorem	cnvco 4940	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 4941*	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 4942*	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 4943*	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 4944*	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 4945*	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 4946*	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 4947*	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 4948	Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
|- dom A = ran `' A
Theorem	dfdmf 4949*	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	csbdmg 4950	Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
|- ( A e. V -> [_ A / x ]_ dom B = dom [_ A / x ]_ B )
Theorem	eldmg 4951*	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 4952*	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 4953*	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 4954*	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 4955	Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
|- ( A C_ B -> dom A C_ dom B )
Theorem	dmeq 4956	Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
|- ( A = B -> dom A = dom B )
Theorem	dmeqi 4957	Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
|- A = B   =>    |- dom A = dom B
Theorem	dmeqd 4958	Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> dom A = dom B )
Theorem	opeldm 4959	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 4960	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	opeldmg 4961	Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)
|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. C -> A e. dom C ) )
Theorem	breldmg 4962	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 4963	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 4964	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 4965	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 4966*	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 4967*	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 4968*	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 4969*	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	dm0 4970	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 4971	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 4972	The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
|- dom _V = _V
Theorem	dm0rn0 4973	An empty domain implies an empty range. For a similar theorem for whether the domain and range are inhabited, see dmmrnm 4976. (Contributed by NM, 21-May-1998.)
|- ( dom A = (/) <-> ran A = (/) )
Theorem	reldm0 4974	A relation is empty iff its domain is empty. For a similar theorem for whether the relation and domain are inhabited, see reldmm 4975. (Contributed by NM, 15-Sep-2004.)
|- ( Rel A -> ( A = (/) <-> dom A = (/) ) )
Theorem	reldmm 4975*	A relation is inhabited iff its domain is inhabited. (Contributed by Jim Kingdon, 30-Jan-2026.)
|- ( Rel A -> ( E. x x e. A <-> E. y y e. dom A ) )
Theorem	dmmrnm 4976*	A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
|- ( E. x x e. dom A <-> E. y y e. ran A )
Theorem	dmxpm 4977*	The domain of a cross 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.)
|- ( E. x x e. B -> dom ( A X. B ) = A )
Theorem	dmxpid 4978	The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.)
|- dom ( A X. A ) = A
Theorem	dmxpin 4979	The domain of the intersection of two square Cartesian products. Unlike dmin 4964, equality holds. (Contributed by NM, 29-Jan-2008.)
|- dom ( ( A X. A ) i^i ( B X. B ) ) = ( A i^i B )
Theorem	xpid11 4980	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 4981	The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5213). (Contributed by NM, 8-Apr-2007.)
|- dom `' `' A = dom A
Theorem	rncnvcnv 4982	The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
|- ran `' `' A = ran A
Theorem	elreldm 4983	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 4984	Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
|- ( A = B -> ran A = ran B )
Theorem	rneqi 4985	Equality inference for range. (Contributed by NM, 4-Mar-2004.)
|- A = B   =>    |- ran A = ran B
Theorem	rneqd 4986	Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> ran A = ran B )
Theorem	rnss 4987	Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
|- ( A C_ B -> ran A C_ ran B )
Theorem	brelrng 4988	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	opelrng 4989	Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)
|- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C )
Theorem	brelrn 4990	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 4991	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 4992	The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
|- ( ( Rel R /\ A R B ) -> A e. dom R )
Theorem	relelrn 4993	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 4994*	Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
|- ( Rel R -> ( A e. dom R <-> E. x A R x ) )
Theorem	relelrnb 4995*	Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
|- ( Rel R -> ( A e. ran R <-> E. x x R A ) )
Theorem	releldmi 4996	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 4997	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 4998*	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 4999*	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 5000*	Membership in a range. (Contributed by NM, 2-Apr-2004.)
|- A e. _V   =>    |- ( A e. ran B <-> E. x x B A )