P. 55 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5401-5500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opelres 5401	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 5402	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	dfres3 5403	Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( A |` B ) = ( A i^i ( B X. ran A ) )
Theorem	opelresg 5404	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 5405	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 5406	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 5407	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 5408	<. 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 5409	The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
|- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )
Theorem	resundi 5410	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 5411	Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
|- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )
Theorem	resindi 5412	Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
|- ( A |` ( B i^i C ) ) = ( ( A |` B ) i^i ( A |` C ) )
Theorem	resindir 5413	Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
|- ( ( A i^i B ) |` C ) = ( ( A |` C ) i^i ( B |` C ) )
Theorem	inres 5414	Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
|- ( A i^i ( B |` C ) ) = ( ( A i^i B ) |` C )
Theorem	resdifcom 5415	Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
|- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )
Theorem	resiun1 5416*	Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.)
|- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )
Theorem	resiun1OLD 5417*	Obsolete proof of resiun1 5416 as of 25-Aug-2021. (Contributed by Mario Carneiro, 29-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )
Theorem	resiun2 5418*	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 5419	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 5420	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 5421	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 5422	A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
|- ( A |` B ) C_ A
Theorem	rescom 5423	Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )
Theorem	ssres 5424	Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
|- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
Theorem	ssres2 5425	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 5426	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 5427	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 5428	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 5429	Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( B C_ C -> ( ( A |` B ) |` C ) = ( A |` B ) )
Theorem	residm 5430	Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
|- ( ( A |` B ) |` B ) = ( A |` B )
Theorem	resima 5431	A restriction to an image. (Contributed by NM, 29-Sep-2004.)
|- ( ( A |` B ) " B ) = ( A " B )
Theorem	resima2 5432	Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)
|- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )
Theorem	resima2OLD 5433	Obsolete proof of resima2 5432 as of 25-Aug-2021. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )
Theorem	xpssres 5434	Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )
Theorem	elres 5435*	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 5436*	Membership 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 5437	Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
|- ( ( Rel A /\ dom A C_ B ) -> ( A |` B ) = A )
Theorem	dmressnsn 5438	The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
|- ( A e. dom F -> dom ( F |` { A } ) = { A } )
Theorem	eldmressnsn 5439	The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
|- ( A e. dom F -> A e. dom ( F |` { A } ) )
Theorem	eldmeldmressn 5440	An element of the domain (of a relation) is an element of the domain of the restriction (of the relation) to the singleton containing this element. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
|- ( X e. dom F <-> X e. dom ( F |` { X } ) )
Theorem	resdm 5441	A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
|- ( Rel A -> ( A |` dom A ) = A )
Theorem	resexg 5442	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 5443	The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
|- A e. _V   =>    |- ( A |` B ) e. _V
Theorem	resindm 5444	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 5445	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 5446*	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	iss 5447	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 5448*	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 5449*	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 5450*	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 5451	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 5452*	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 5453*	Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) }
Theorem	mptss 5454*	Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( A C_ B -> ( x e. A |-> C ) C_ ( x e. B |-> C ) )
Theorem	opabresid 5455*	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 5456*	The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
|- ( x e. A |-> x ) = ( _I |` A )
Theorem	dmresi 5457	The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
|- dom ( _I |` A ) = A
Theorem	restidsing 5458	Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)
|- ( _I |` { A } ) = ( { A } X. { A } )
Theorem	restidsingOLD 5459	Obsolete proof of restidsing 5458 as of 25-Aug-2021. (Contributed by FL, 2-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( _I |` { A } ) = ( { A } X. { A } )
Theorem	resid 5460	Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
|- ( Rel A -> ( A |` _V ) = A )
Theorem	imaeq1 5461	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
|- ( A = B -> ( A " C ) = ( B " C ) )
Theorem	imaeq2 5462	Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
|- ( A = B -> ( C " A ) = ( C " B ) )
Theorem	imaeq1i 5463	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( A " C ) = ( B " C )
Theorem	imaeq2i 5464	Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( C " A ) = ( C " B )
Theorem	imaeq1d 5465	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A " C ) = ( B " C ) )
Theorem	imaeq2d 5466	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C " A ) = ( C " B ) )
Theorem	imaeq12d 5467	Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A " C ) = ( B " D ) )
Theorem	dfima2 5468*	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 5469*	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 5470*	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 5471*	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 5472*	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 5473*	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 5474	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 5475	Deduction version of bound-variable hypothesis builder nfima 5474. (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 5476	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 5477	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	imai 5478	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
|- ( _I " A ) = A
Theorem	rnresi 5479	The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
|- ran ( _I |` A ) = A
Theorem	resiima 5480	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
|- ( B C_ A -> ( ( _I |` A ) " B ) = B )
Theorem	ima0 5481	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
|- ( A " (/) ) = (/)
Theorem	0ima 5482	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
|- ( (/) " A ) = (/)
Theorem	csbima12 5483	Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.)
|- [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B )
Theorem	imadisj 5484	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 5485	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 5486	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 5487*	The image of a singleton. (Contributed by NM, 8-May-2005.)
|- ( A e. B -> ( R " { A } ) = { y | A R y } )
Theorem	relimasn 5488*	The image of a singleton. (Contributed by NM, 20-May-1998.)
|- ( Rel R -> ( R " { A } ) = { y | A R y } )
Theorem	elrelimasn 5489	Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
|- ( Rel R -> ( B e. ( R " { A } ) <-> A R B ) )
Theorem	elimasn 5490	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 5491	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	elimasni 5492	Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)
|- ( C e. ( A " { B } ) -> B A C )
Theorem	args 5493*	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). Observe the resemblance to the alternate definition dffv4 6188 of function value, which is based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)
|- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y }
Theorem	eliniseg 5494	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 5495	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 5496*	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	inisegn0 5497	Nonemptiness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)
|- ( A e. ran F <-> ( `' F " { A } ) =/= (/) )
Theorem	dffr3 5498*	Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
|- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) )
Theorem	dfse2 5499*	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	imass1 5500	Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
|- ( A C_ B -> ( A " C ) C_ ( B " C ) )