P. 49 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4801-4900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rnex 4801	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 4802	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 4803	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 4804	Range of a composition. (Contributed by NM, 19-Mar-1998.)
|- ran ( A o. B ) C_ ran A
Theorem	dmcosseq 4805	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 4806	Domain of a composition. (Contributed by NM, 19-Mar-1998.)
|- ( dom A = ran B -> dom ( A o. B ) = dom B )
Theorem	rncoeq 4807	Range of a composition. (Contributed by NM, 19-Mar-1998.)
|- ( dom A = ran B -> ran ( A o. B ) = ran A )
Theorem	reseq1 4808	Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
|- ( A = B -> ( A |` C ) = ( B |` C ) )
Theorem	reseq2 4809	Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
|- ( A = B -> ( C |` A ) = ( C |` B ) )
Theorem	reseq1i 4810	Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
|- A = B   =>    |- ( A |` C ) = ( B |` C )
Theorem	reseq2i 4811	Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = B   =>    |- ( C |` A ) = ( C |` B )
Theorem	reseq12i 4812	Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
|- A = B   &    |- C = D   =>    |- ( A |` C ) = ( B |` D )
Theorem	reseq1d 4813	Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A |` C ) = ( B |` C ) )
Theorem	reseq2d 4814	Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C |` A ) = ( C |` B ) )
Theorem	reseq12d 4815	Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A |` C ) = ( B |` D ) )
Theorem	nfres 4816	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 4817	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 4818	A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
|- ( A |` (/) ) = (/)
Theorem	opelres 4819	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 4820	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 4821	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 4822	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 4823	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 4824	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 4825	<. 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 4826	The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
|- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )
Theorem	resundi 4827	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 4828	Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
|- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
Theorem	resindi 4829	Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
|- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )
Theorem	resindir 4830	Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
|- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
Theorem	inres 4831	Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
|- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C )
Theorem	resdifcom 4832	Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
|- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )
Theorem	resiun1 4833*	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 4834*	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 4835	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 4836	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 4837	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 4838	A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
|- ( A |` B ) C_ A
Theorem	rescom 4839	Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
Theorem	ssres 4840	Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
|- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
Theorem	ssres2 4841	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 4842	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 4843	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 4844	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 4845	Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )
Theorem	residm 4846	Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( ( A |` B ) |` B ) = ( A |` B )
Theorem	resima 4847	A restriction to an image. (Contributed by NM, 29-Sep-2004.)
|- ( ( A |` B ) " B ) = ( A " B )
Theorem	resima2 4848	Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
|- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )
Theorem	xpssres 4849	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 4850*	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 4851*	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 4852	Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
|- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )
Theorem	resdm 4853	A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
|- ( Rel A -> ( A |` dom A ) = A )
Theorem	resexg 4854	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 4855	The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
|- A e. _V   =>    |- ( A |` B ) e. _V
Theorem	resindm 4856	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 4857	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 4858*	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 4859	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 4860	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 4861*	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 4862*	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 4863*	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 4864	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 4865*	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 4866*	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 4867*	The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
|- { <. x , y >. | ( x e. A /\ y = x ) } = ( _I |` A )
Theorem	mptresid 4868*	The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)
|- ( x e. A |-> x ) = ( _I |` A )
Theorem	dmresi 4869	The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
|- dom ( _I |` A ) = A
Theorem	resid 4870	Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
|- ( Rel A -> ( A |` _V ) = A )
Theorem	imaeq1 4871	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
|- ( A = B -> ( A " C ) = ( B " C ) )
Theorem	imaeq2 4872	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
|- ( A = B -> ( C " A ) = ( C " B ) )
Theorem	imaeq1i 4873	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( A " C ) = ( B " C )
Theorem	imaeq2i 4874	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( C " A ) = ( C " B )
Theorem	imaeq1d 4875	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A " C ) = ( B " C ) )
Theorem	imaeq2d 4876	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C " A ) = ( C " B ) )
Theorem	imaeq12d 4877	Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A " C ) = ( B " D ) )
Theorem	dfima2 4878*	Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A " B ) = { y | E. x e. B x A y }
Theorem	dfima3 4879*	Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A " B ) = { y | E. x ( x e. B /\ <. x , y >. e. A ) }
Theorem	elimag 4880*	Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
|- ( A e. V -> ( A e. ( B " C ) <-> E. x e. C x B A ) )
Theorem	elima 4881*	Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
|- A e. _V   =>    |- ( A e. ( B " C ) <-> E. x e. C x B A )
Theorem	elima2 4882*	Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
|- A e. _V   =>    |- ( A e. ( B " C ) <-> E. x ( x e. C /\ x B A ) )
Theorem	elima3 4883*	Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
|- A e. _V   =>    |- ( A e. ( B " C ) <-> E. x ( x e. C /\ <. x , A >. e. B ) )
Theorem	nfima 4884	Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A " B )
Theorem	nfimad 4885	Deduction version of bound-variable hypothesis builder nfima 4884. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x ( A " B ) )
Theorem	imadmrn 4886	The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)
|- ( A " dom A ) = ran A
Theorem	imassrn 4887	The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)
|- ( A " B ) C_ ran A
Theorem	imaexg 4888	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
|- ( A e. V -> ( A " B ) e. _V )
Theorem	imaex 4889	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)
|- A e. _V   =>    |- ( A " B ) e. _V
Theorem	imai 4890	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
|- ( _I " A ) = A
Theorem	rnresi 4891	The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
|- ran ( _I |` A ) = A
Theorem	resiima 4892	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
|- ( B C_ A -> ( ( _I |` A ) " B ) = B )
Theorem	ima0 4893	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
|- ( A " (/) ) = (/)
Theorem	0ima 4894	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
|- ( (/) " A ) = (/)
Theorem	csbima12g 4895	Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
|- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
Theorem	imadisj 4896	A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
|- ( ( A " B ) = (/) <-> ( dom A i^i B ) = (/) )
Theorem	cnvimass 4897	A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
|- ( `' A " B ) C_ dom A
Theorem	cnvimarndm 4898	The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)
|- ( `' A " ran A ) = dom A
Theorem	imasng 4899*	The image of a singleton. (Contributed by NM, 8-May-2005.)
|- ( A e. B -> ( R " { A } ) = { y | A R y } )
Theorem	elreimasng 4900	Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
|- ( ( Rel R /\ A e. V ) -> ( B e. ( R " { A } ) <-> A R B ) )