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	resieq 4901	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 4902	<. 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 4903	The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
|- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )
Theorem	resundi 4904	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 4905	Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
|- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
Theorem	resindi 4906	Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
|- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )
Theorem	resindir 4907	Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
|- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
Theorem	inres 4908	Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
|- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C )
Theorem	resdifcom 4909	Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
|- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )
Theorem	resiun1 4910*	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 4911*	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 4912	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 4913	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 4914	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 4915	A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
|- ( A |` B ) C_ A
Theorem	rescom 4916	Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
Theorem	ssres 4917	Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
|- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
Theorem	ssres2 4918	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 4919	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 4920	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 4921	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 4922	Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )
Theorem	residm 4923	Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( ( A |` B ) |` B ) = ( A |` B )
Theorem	resima 4924	A restriction to an image. (Contributed by NM, 29-Sep-2004.)
|- ( ( A |` B ) " B ) = ( A " B )
Theorem	resima2 4925	Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
|- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )
Theorem	xpssres 4926	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 4927*	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 4928*	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 4929	Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
|- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )
Theorem	resdm 4930	A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
|- ( Rel A -> ( A |` dom A ) = A )
Theorem	resexg 4931	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 4932	The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
|- A e. _V   =>    |- ( A |` B ) e. _V
Theorem	resindm 4933	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 4934	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 4935*	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 4936	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 4937	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 4938*	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 4939*	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 4940*	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 4941	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 4942*	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 4943*	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 4944*	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 4945*	The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)
|- ( x e. A |-> x ) = ( _I |` A )
Theorem	dmresi 4946	The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
|- dom ( _I |` A ) = A
Theorem	resid 4947	Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
|- ( Rel A -> ( A |` _V ) = A )
Theorem	imaeq1 4948	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
|- ( A = B -> ( A " C ) = ( B " C ) )
Theorem	imaeq2 4949	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
|- ( A = B -> ( C " A ) = ( C " B ) )
Theorem	imaeq1i 4950	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( A " C ) = ( B " C )
Theorem	imaeq2i 4951	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( C " A ) = ( C " B )
Theorem	imaeq1d 4952	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A " C ) = ( B " C ) )
Theorem	imaeq2d 4953	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C " A ) = ( C " B ) )
Theorem	imaeq12d 4954	Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A " C ) = ( B " D ) )
Theorem	dfima2 4955*	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 4956*	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 4957*	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 4958*	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 4959*	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 4960*	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 4961	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 4962	Deduction version of bound-variable hypothesis builder nfima 4961. (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 4963	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 4964	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 4965	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 4966	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 4967	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
|- ( _I " A ) = A
Theorem	rnresi 4968	The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
|- ran ( _I |` A ) = A
Theorem	resiima 4969	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
|- ( B C_ A -> ( ( _I |` A ) " B ) = B )
Theorem	ima0 4970	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
|- ( A " (/) ) = (/)
Theorem	0ima 4971	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
|- ( (/) " A ) = (/)
Theorem	csbima12g 4972	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 4973	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 4974	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 4975	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 4976*	The image of a singleton. (Contributed by NM, 8-May-2005.)
|- ( A e. B -> ( R " { A } ) = { y | A R y } )
Theorem	elreimasng 4977	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 ) )
Theorem	elimasn 4978	Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- B e. _V   &    |- C e. _V   =>    |- ( C e. ( A " { B } ) <-> <. B , C >. e. A )
Theorem	elimasng 4979	Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
|- ( ( B e. V /\ C e. W ) -> ( C e. ( A " { B } ) <-> <. B , C >. e. A ) )
Theorem	args 4980*	Two ways to express the class of unique-valued arguments of F, which is the same as the domain of F whenever F is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg F " for this class (for which we have no separate notation). (Contributed by NM, 8-May-2005.)
|- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y }
Theorem	eliniseg 4981	Membership in an initial segment. The idiom ( `' A " { B } ), meaning { x | x A B }, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- C e. _V   =>    |- ( B e. V -> ( C e. ( `' A " { B } ) <-> C A B ) )
Theorem	epini 4982	Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
|- A e. _V   =>    |- ( `' _E " { A } ) = A
Theorem	iniseg 4983*	An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.)
|- ( B e. V -> ( `' A " { B } ) = { x | x A B } )
Theorem	dfse2 4984*	Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
|- ( R Se A <-> A. x e. A ( A i^i ( `' R " { x } ) ) e. _V )
Theorem	exse2 4985	Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
|- ( R e. V -> R Se A )
Theorem	imass1 4986	Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
|- ( A C_ B -> ( A " C ) C_ ( B " C ) )
Theorem	imass2 4987	Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
|- ( A C_ B -> ( C " A ) C_ ( C " B ) )
Theorem	ndmima 4988	The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
|- ( -. A e. dom B -> ( B " { A } ) = (/) )
Theorem	relcnv 4989	A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
|- Rel `' A
Theorem	relbrcnvg 4990	When R is a relation, the sethood assumptions on brcnv 4794 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- ( Rel R -> ( A `' R B <-> B R A ) )
Theorem	relbrcnv 4991	When R is a relation, the sethood assumptions on brcnv 4794 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- Rel R   =>    |- ( A `' R B <-> B R A )
Theorem	cotr 4992*	Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( R o. R ) C_ R <-> A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) )
Theorem	issref 4993*	Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
|- ( ( _I |` A ) C_ R <-> A. x e. A x R x )
Theorem	cnvsym 4994*	Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) )
Theorem	intasym 4995*	Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( R i^i `' R ) C_ _I <-> A. x A. y ( ( x R y /\ y R x ) -> x = y ) )
Theorem	asymref 4996*	Two ways of saying a relation is antisymmetric and reflexive. U. U. R is the field of a relation by relfld 5139. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( R i^i `' R ) = ( _I |` U. U. R ) <-> A. x e. U. U. R A. y ( ( x R y /\ y R x ) <-> x = y ) )
Theorem	intirr 4997*	Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( R i^i _I ) = (/) <-> A. x -. x R x )
Theorem	brcodir 4998*	Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
|- ( ( A e. V /\ B e. W ) -> ( A ( `' R o. R ) B <-> E. z ( A R z /\ B R z ) ) )
Theorem	codir 4999*	Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)
|- ( ( A X. B ) C_ ( `' R o. R ) <-> A. x e. A A. y e. B E. z ( x R z /\ y R z ) )
Theorem	qfto 5000*	A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)
|- ( ( A X. B ) C_ ( R u. `' R ) <-> A. x e. A A. y e. B ( x R y \/ y R x ) )