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	opelrn 4901	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 4902	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 4903	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 4904*	Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
|- ( Rel R -> ( A e. dom R <-> E. x A R x ) )
Theorem	relelrnb 4905*	Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
|- ( Rel R -> ( A e. ran R <-> E. x x R A ) )
Theorem	releldmi 4906	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 4907	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 4908*	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 4909*	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 4910*	Membership in a range. (Contributed by NM, 2-Apr-2004.)
|- A e. _V   =>    |- ( A e. ran B <-> E. x x B A )
Theorem	nfdm 4911	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 4912	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 4913	Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
|- dom |^|_ x e. A B C_ |^|_ x e. A dom B
Theorem	rnopab 4914*	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 4915*	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 4916*	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 4917*	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 4918	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 4919*	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 4920*	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 4921*	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 4922*	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 4923	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 4924	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 4925	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 4926	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 4927	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 4928*	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 4929	A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
|- ( Rel A -> ( A = (/) <-> ran A = (/) ) )
Theorem	dmrnssfld 4930	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	dmexg 4931	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
|- ( A e. V -> dom A e. _V )
Theorem	rnexg 4932	The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
|- ( A e. V -> ran A e. _V )
Theorem	dmex 4933	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
|- A e. _V   =>    |- dom A e. _V
Theorem	rnex 4934	The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
|- A e. _V   =>    |- ran A e. _V
Theorem	iprc 4935	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. (Contributed by NM, 1-Jan-2007.)
|- -. _I e. _V
Theorem	dmcoss 4936	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 4937	Range of a composition. (Contributed by NM, 19-Mar-1998.)
|- ran ( A o. B ) C_ ran A
Theorem	dmcosseq 4938	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 4939	Domain of a composition. (Contributed by NM, 19-Mar-1998.)
|- ( dom A = ran B -> dom ( A o. B ) = dom B )
Theorem	rncoeq 4940	Range of a composition. (Contributed by NM, 19-Mar-1998.)
|- ( dom A = ran B -> ran ( A o. B ) = ran A )
Theorem	reseq1 4941	Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
|- ( A = B -> ( A |` C ) = ( B |` C ) )
Theorem	reseq2 4942	Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
|- ( A = B -> ( C |` A ) = ( C |` B ) )
Theorem	reseq1i 4943	Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
|- A = B   =>    |- ( A |` C ) = ( B |` C )
Theorem	reseq2i 4944	Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = B   =>    |- ( C |` A ) = ( C |` B )
Theorem	reseq12i 4945	Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
|- A = B   &    |- C = D   =>    |- ( A |` C ) = ( B |` D )
Theorem	reseq1d 4946	Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A |` C ) = ( B |` C ) )
Theorem	reseq2d 4947	Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C |` A ) = ( C |` B ) )
Theorem	reseq12d 4948	Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A |` C ) = ( B |` D ) )
Theorem	nfres 4949	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	csbresg 4950	Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) )
Theorem	res0 4951	A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
|- ( A |` (/) ) = (/)
Theorem	opelres 4952	Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
|- B e. _V   =>    |- ( <. A , B >. e. ( C |` D ) <-> ( <. A , B >. e. C /\ A e. D ) )
Theorem	brres 4953	Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
|- B e. _V   =>    |- ( A ( C |` D ) B <-> ( A C B /\ A e. D ) )
Theorem	opelresg 4954	Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
|- ( B e. V -> ( <. A , B >. e. ( C |` D ) <-> ( <. A , B >. e. C /\ A e. D ) ) )
Theorem	brresg 4955	Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
|- ( B e. V -> ( A ( C |` D ) B <-> ( A C B /\ A e. D ) ) )
Theorem	opres 4956	Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- B e. _V   =>    |- ( A e. D -> ( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) )
Theorem	resieq 4957	A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
|- ( ( B e. A /\ C e. A ) -> ( B ( _I |` A ) C <-> B = C ) )
Theorem	opelresi 4958	<. A , A >. belongs to a restriction of the identity class iff A belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)
|- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> A e. B ) )
Theorem	resres 4959	The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
|- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )
Theorem	resundi 4960	Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
|- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )
Theorem	resundir 4961	Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
|- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
Theorem	resindi 4962	Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
|- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )
Theorem	resindir 4963	Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
|- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
Theorem	inres 4964	Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
|- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C )
Theorem	resdifcom 4965	Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
|- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )
Theorem	resiun1 4966*	Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
|- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )
Theorem	resiun2 4967*	Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
|- ( C |` U_ x e. A B ) = U_ x e. A ( C |` B )
Theorem	dmres 4968	The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
|- dom ( A |` B ) = ( B i^i dom A )
Theorem	ssdmres 4969	A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
|- ( A C_ dom B <-> dom ( B |` A ) = A )
Theorem	dmresexg 4970	The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
|- ( B e. V -> dom ( A |` B ) e. _V )
Theorem	resss 4971	A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
|- ( A |` B ) C_ A
Theorem	rescom 4972	Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
Theorem	ssres 4973	Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
|- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
Theorem	ssres2 4974	Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )
Theorem	relres 4975	A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- Rel ( A |` B )
Theorem	resabs1 4976	Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
|- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )
Theorem	resabs1d 4977	Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> B C_ C )   =>    |- ( ph -> ( ( A |` C ) |` B ) = ( A |` B ) )
Theorem	resabs2 4978	Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )
Theorem	residm 4979	Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( ( A |` B ) |` B ) = ( A |` B )
Theorem	resima 4980	A restriction to an image. (Contributed by NM, 29-Sep-2004.)
|- ( ( A |` B ) " B ) = ( A " B )
Theorem	resima2 4981	Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
|- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )
Theorem	xpssres 4982	Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )
Theorem	elres 4983*	Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
|- ( A e. ( B |` C ) <-> E. x e. C E. y ( A = <. x , y >. /\ <. x , y >. e. B ) )
Theorem	elsnres 4984*	Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
|- C e. _V   =>    |- ( A e. ( B |` { C } ) <-> E. y ( A = <. C , y >. /\ <. C , y >. e. B ) )
Theorem	relssres 4985	Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
|- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )
Theorem	resdm 4986	A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
|- ( Rel A -> ( A |` dom A ) = A )
Theorem	resexg 4987	The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A e. V -> ( A |` B ) e. _V )
Theorem	resex 4988	The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
|- A e. _V   =>    |- ( A |` B ) e. _V
Theorem	resindm 4989	When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)
|- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )
Theorem	resdmdfsn 4990	Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (Contributed by AV, 2-Dec-2018.)
|- ( Rel R -> ( R |` ( _V \ { X } ) ) = ( R |` ( dom R \ { X } ) ) )
Theorem	resopab 4991*	Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
|- ( { <. x , y >. | ph } |` A ) = { <. x , y >. | ( x e. A /\ ph ) }
Theorem	resiexg 4992	The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
|- ( A e. V -> ( _I |` A ) e. _V )
Theorem	iss 4993	A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A C_ _I <-> A = ( _I |` dom A ) )
Theorem	resopab2 4994*	Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
|- ( A C_ B -> ( { <. x , y >. | ( x e. B /\ ph ) } |` A ) = { <. x , y >. | ( x e. A /\ ph ) } )
Theorem	resmpt 4995*	Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
|- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )
Theorem	resmpt3 4996*	Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
|- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C )
Theorem	resmptf 4997	Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )
Theorem	resmptd 4998*	Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> B C_ A )   =>    |- ( ph -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )
Theorem	dfres2 4999*	Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) }
Theorem	opabresid 5000*	The restricted identity relation expressed as an ordered-pair class abstraction. (Contributed by FL, 25-Apr-2012.)
|- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }