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	nfco 4901	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 4902*	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 4903*	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 4904*	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 4905	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 4906*	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 4907*	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 4908*	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 4909	Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
|- ( A C_ B -> `' A C_ `' B )
Theorem	cnveq 4910	Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
|- ( A = B -> `' A = `' B )
Theorem	cnveqi 4911	Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
|- A = B   =>    |- `' A = `' B
Theorem	cnveqd 4912	Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> `' A = `' B )
Theorem	elcnv 4913*	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 4914*	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 4915	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 4916	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 4917	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 4918	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 4919	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 4920	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 4921	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 4922*	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 4923*	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 4924*	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 4925*	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 4926*	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 4927*	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 4928*	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 4929	Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
|- dom A = ran `' A
Theorem	dfdmf 4930*	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 4931	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 4932*	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 4933*	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 4934*	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 4935*	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 4936	Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
|- ( A C_ B -> dom A C_ dom B )
Theorem	dmeq 4937	Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
|- ( A = B -> dom A = dom B )
Theorem	dmeqi 4938	Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
|- A = B   =>    |- dom A = dom B
Theorem	dmeqd 4939	Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> dom A = dom B )
Theorem	opeldm 4940	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 4941	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 4942	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 4943	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 4944	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 4945	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 4946	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 4947*	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 4948*	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 4949*	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 4950*	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 4951	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 4952	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 4953	The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
|- dom _V = _V
Theorem	dm0rn0 4954	An empty domain implies an empty range. For a similar theorem for whether the domain and range are inhabited, see dmmrnm 4957. (Contributed by NM, 21-May-1998.)
|- ( dom A = (/) <-> ran A = (/) )
Theorem	reldm0 4955	A relation is empty iff its domain is empty. For a similar theorem for whether the relation and domain are inhabited, see reldmm 4956. (Contributed by NM, 15-Sep-2004.)
|- ( Rel A -> ( A = (/) <-> dom A = (/) ) )
Theorem	reldmm 4956*	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 4957*	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 4958*	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 4959	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 4960	The domain of the intersection of two square Cartesian products. Unlike dmin 4945, equality holds. (Contributed by NM, 29-Jan-2008.)
|- dom ( ( A X. A ) i^i ( B X. B ) ) = ( A i^i B )
Theorem	xpid11 4961	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 4962	The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5194). (Contributed by NM, 8-Apr-2007.)
|- dom `' `' A = dom A
Theorem	rncnvcnv 4963	The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
|- ran `' `' A = ran A
Theorem	elreldm 4964	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 4965	Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
|- ( A = B -> ran A = ran B )
Theorem	rneqi 4966	Equality inference for range. (Contributed by NM, 4-Mar-2004.)
|- A = B   =>    |- ran A = ran B
Theorem	rneqd 4967	Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> ran A = ran B )
Theorem	rnss 4968	Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
|- ( A C_ B -> ran A C_ ran B )
Theorem	brelrng 4969	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 4970	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 4971	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 4972	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 4973	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 4974	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 4975*	Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
|- ( Rel R -> ( A e. dom R <-> E. x A R x ) )
Theorem	relelrnb 4976*	Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
|- ( Rel R -> ( A e. ran R <-> E. x x R A ) )
Theorem	releldmi 4977	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 4978	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 4979*	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 4980*	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 4981*	Membership in a range. (Contributed by NM, 2-Apr-2004.)
|- A e. _V   =>    |- ( A e. ran B <-> E. x x B A )
Theorem	nfdm 4982	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 4983	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 4984	Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
|- dom |^|_ x e. A B C_ |^|_ x e. A dom B
Theorem	rnopab 4985*	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 4986*	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 4987*	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 4988*	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 4989	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 4990*	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 4991*	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	elrnmptdv 4992*	Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- F = ( x e. A |-> B )   &    |- ( ph -> C e. A )   &    |- ( ph -> D e. V )   &    |- ( ( ph /\ x = C ) -> D = B )   =>    |- ( ph -> D e. ran F )
Theorem	elrnmpt2d 4993*	Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- F = ( x e. A |-> B )   &    |- ( ph -> C e. ran F )   =>    |- ( ph -> E. x e. A C = B )
Theorem	rn0 4994	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 4995	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 4996	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 4997	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 4998	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 4999*	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 5000	A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
|- ( Rel A -> ( A = (/) <-> ran A = (/) ) )