P. 52 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rnxpid 5101	The range of a square cross product. (Contributed by FL, 17-May-2010.)
|- ran ( A X. A ) = A
Theorem	ssxpbm 5102*	A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)
|- ( E. x x e. ( A X. B ) -> ( ( A X. B ) C_ ( C X. D ) <-> ( A C_ C /\ B C_ D ) ) )
Theorem	ssxp1 5103*	Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
|- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )
Theorem	ssxp2 5104*	Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
|- ( E. x x e. C -> ( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) )
Theorem	xp11m 5105*	The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)
|- ( ( E. x x e. A /\ E. y y e. B ) -> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) )
Theorem	xpcanm 5106*	Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
|- ( E. x x e. C -> ( ( C X. A ) = ( C X. B ) <-> A = B ) )
Theorem	xpcan2m 5107*	Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
|- ( E. x x e. C -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )
Theorem	xpexr2m 5108*	If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)
|- ( ( ( A X. B ) e. C /\ E. x x e. ( A X. B ) ) -> ( A e. _V /\ B e. _V ) )
Theorem	ssrnres 5109	Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
|- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )
Theorem	rninxp 5110*	Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ran ( C i^i ( A X. B ) ) = B <-> A. y e. B E. x e. A x C y )
Theorem	dminxp 5111*	Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)
|- ( dom ( C i^i ( A X. B ) ) = A <-> A. x e. A E. y e. B x C y )
Theorem	imainrect 5112	Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)
|- ( ( G i^i ( A X. B ) ) " Y ) = ( ( G " ( Y i^i A ) ) i^i B )
Theorem	xpima1 5113	The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
|- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )
Theorem	xpima2m 5114*	The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
|- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B )
Theorem	xpimasn 5115	The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)
|- ( X e. A -> ( ( A X. B ) " { X } ) = B )
Theorem	cnvcnv3 5116*	The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)
|- `' `' R = { <. x , y >. | x R y }
Theorem	dfrel2 5117	Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
|- ( Rel R <-> `' `' R = R )
Theorem	dfrel4v 5118*	A relation can be expressed as the set of ordered pairs in it. (Contributed by Mario Carneiro, 16-Aug-2015.)
|- ( Rel R <-> R = { <. x , y >. | x R y } )
Theorem	cnvcnv 5119	The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)
|- `' `' A = ( A i^i ( _V X. _V ) )
Theorem	cnvcnv2 5120	The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
|- `' `' A = ( A |` _V )
Theorem	cnvcnvss 5121	The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
|- `' `' A C_ A
Theorem	cnveqb 5122	Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
|- ( ( Rel A /\ Rel B ) -> ( A = B <-> `' A = `' B ) )
Theorem	cnveq0 5123	A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
|- ( Rel A -> ( A = (/) <-> `' A = (/) ) )
Theorem	dfrel3 5124	Alternate definition of relation. (Contributed by NM, 14-May-2008.)
|- ( Rel R <-> ( R |` _V ) = R )
Theorem	dmresv 5125	The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
|- dom ( A |` _V ) = dom A
Theorem	rnresv 5126	The range of a universal restriction. (Contributed by NM, 14-May-2008.)
|- ran ( A |` _V ) = ran A
Theorem	dfrn4 5127	Range defined in terms of image. (Contributed by NM, 14-May-2008.)
|- ran A = ( A " _V )
Theorem	csbrng 5128	Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> [_ A / x ]_ ran B = ran [_ A / x ]_ B )
Theorem	rescnvcnv 5129	The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( `' `' A |` B ) = ( A |` B )
Theorem	cnvcnvres 5130	The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
|- `' `' ( A |` B ) = ( `' `' A |` B )
Theorem	imacnvcnv 5131	The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
|- ( `' `' A " B ) = ( A " B )
Theorem	dmsnm 5132*	The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
|- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } )
Theorem	rnsnm 5133*	The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
|- ( A e. ( _V X. _V ) <-> E. x x e. ran { A } )
Theorem	dmsn0 5134	The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
|- dom { (/) } = (/)
Theorem	cnvsn0 5135	The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- `' { (/) } = (/)
Theorem	dmsn0el 5136	The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
|- ( (/) e. A -> dom { A } = (/) )
Theorem	relsn2m 5137*	A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)
|- A e. _V   =>    |- ( Rel { A } <-> E. x x e. dom { A } )
Theorem	dmsnopg 5138	The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( B e. V -> dom { <. A , B >. } = { A } )
Theorem	dmpropg 5139	The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ( B e. V /\ D e. W ) -> dom { <. A , B >. , <. C , D >. } = { A , C } )
Theorem	dmsnop 5140	The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- B e. _V   =>    |- dom { <. A , B >. } = { A }
Theorem	dmprop 5141	The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)
|- B e. _V   &    |- D e. _V   =>    |- dom { <. A , B >. , <. C , D >. } = { A , C }
Theorem	dmtpop 5142	The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)
|- B e. _V   &    |- D e. _V   &    |- F e. _V   =>    |- dom { <. A , B >. , <. C , D >. , <. E , F >. } = { A , C , E }
Theorem	cnvcnvsn 5143	Double converse of a singleton of an ordered pair. (Unlike cnvsn 5149, this does not need any sethood assumptions on A and B.) (Contributed by Mario Carneiro, 26-Apr-2015.)
|- `' `' { <. A , B >. } = `' { <. B , A >. }
Theorem	dmsnsnsng 5144	The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
|- ( A e. _V -> dom { { { A } } } = { A } )
Theorem	rnsnopg 5145	The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( A e. V -> ran { <. A , B >. } = { B } )
Theorem	rnpropg 5146	The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
|- ( ( A e. V /\ B e. W ) -> ran { <. A , C >. , <. B , D >. } = { C , D } )
Theorem	rnsnop 5147	The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   =>    |- ran { <. A , B >. } = { B }
Theorem	op1sta 5148	Extract the first member of an ordered pair. (See op2nda 5151 to extract the second member and op1stb 4510 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- U. dom { <. A , B >. } = A
Theorem	cnvsn 5149	Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- `' { <. A , B >. } = { <. B , A >. }
Theorem	op2ndb 5150	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4510 to extract the first member and op2nda 5151 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
|- A e. _V   &    |- B e. _V   =>    |- |^| |^| |^| `' { <. A , B >. } = B
Theorem	op2nda 5151	Extract the second member of an ordered pair. (See op1sta 5148 to extract the first member and op2ndb 5150 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- A e. _V   &    |- B e. _V   =>    |- U. ran { <. A , B >. } = B
Theorem	cnvsng 5152	Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
|- ( ( A e. V /\ B e. W ) -> `' { <. A , B >. } = { <. B , A >. } )
Theorem	opswapg 5153	Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
|- ( ( A e. V /\ B e. W ) -> U. `' { <. A , B >. } = <. B , A >. )
Theorem	elxp4 5154	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5155. (Contributed by NM, 17-Feb-2004.)
|- ( A e. ( B X. C ) <-> ( A = <. U. dom { A } , U. ran { A } >. /\ ( U. dom { A } e. B /\ U. ran { A } e. C ) ) )
Theorem	elxp5 5155	Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5154 when the double intersection does not create class existence problems (caused by int0 3885). (Contributed by NM, 1-Aug-2004.)
|- ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )
Theorem	cnvresima 5156	An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
|- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A )
Theorem	resdm2 5157	A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
|- ( A |` dom A ) = `' `' A
Theorem	resdmres 5158	Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
|- ( A |` dom ( A |` B ) ) = ( A |` B )
Theorem	imadmres 5159	The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
|- ( A " dom ( A |` B ) ) = ( A " B )
Theorem	mptpreima 5160*	The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
|- F = ( x e. A |-> B )   =>    |- ( `' F " C ) = { x e. A | B e. C }
Theorem	mptiniseg 5161*	Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
|- F = ( x e. A |-> B )   =>    |- ( C e. V -> ( `' F " { C } ) = { x e. A | B = C } )
Theorem	dmmpt 5162	The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
|- F = ( x e. A |-> B )   =>    |- dom F = { x e. A | B e. _V }
Theorem	dmmptss 5163*	The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
|- F = ( x e. A |-> B )   =>    |- dom F C_ A
Theorem	dmmptg 5164*	The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
|- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A )
Theorem	relco 5165	A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
|- Rel ( A o. B )
Theorem	dfco2 5166*	Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
|- ( A o. B ) = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )
Theorem	dfco2a 5167*	Generalization of dfco2 5166, where C can have any value between dom A i^i ran B and _V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( dom A i^i ran B ) C_ C -> ( A o. B ) = U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) )
Theorem	coundi 5168	Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A o. ( B u. C ) ) = ( ( A o. B ) u. ( A o. C ) )
Theorem	coundir 5169	Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( A u. B ) o. C ) = ( ( A o. C ) u. ( B o. C ) )
Theorem	cores 5170	Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) )
Theorem	resco 5171	Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
|- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )
Theorem	imaco 5172	Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
|- ( ( A o. B ) " C ) = ( A " ( B " C ) )
Theorem	rnco 5173	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
|- ran ( A o. B ) = ran ( A |` ran B )
Theorem	rnco2 5174	The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
|- ran ( A o. B ) = ( A " ran B )
Theorem	dmco 5175	The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
|- dom ( A o. B ) = ( `' B " dom A )
Theorem	coiun 5176*	Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
|- ( A o. U_ x e. C B ) = U_ x e. C ( A o. B )
Theorem	cocnvcnv1 5177	A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
|- ( `' `' A o. B ) = ( A o. B )
Theorem	cocnvcnv2 5178	A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
|- ( A o. `' `' B ) = ( A o. B )
Theorem	cores2 5179	Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
|- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )
Theorem	co02 5180	Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
|- ( A o. (/) ) = (/)
Theorem	co01 5181	Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
|- ( (/) o. A ) = (/)
Theorem	coi1 5182	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
|- ( Rel A -> ( A o. _I ) = A )
Theorem	coi2 5183	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
|- ( Rel A -> ( _I o. A ) = A )
Theorem	coires1 5184	Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
|- ( A o. ( _I |` B ) ) = ( A |` B )
Theorem	coass 5185	Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
|- ( ( A o. B ) o. C ) = ( A o. ( B o. C ) )
Theorem	relcnvtr 5186	A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
|- ( Rel R -> ( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) )
Theorem	relssdmrn 5187	A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
|- ( Rel A -> A C_ ( dom A X. ran A ) )
Theorem	cnvssrndm 5188	The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- `' A C_ ( ran A X. dom A )
Theorem	cossxp 5189	Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
|- ( A o. B ) C_ ( dom B X. ran A )
Theorem	cossxp2 5190	The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> R C_ ( A X. B ) )   &    |- ( ph -> S C_ ( B X. C ) )   =>    |- ( ph -> ( S o. R ) C_ ( A X. C ) )
Theorem	cocnvres 5191	Restricting a relation and a converse relation when they are composed together. (Contributed by BJ, 10-Jul-2022.)
|- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
Theorem	cocnvss 5192	Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)
|- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
Theorem	relrelss 5193	Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
|- ( ( Rel A /\ Rel dom A ) <-> A C_ ( ( _V X. _V ) X. _V ) )
Theorem	unielrel 5194	The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
|- ( ( Rel R /\ A e. R ) -> U. A e. U. R )
Theorem	relfld 5195	The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
|- ( Rel R -> U. U. R = ( dom R u. ran R ) )
Theorem	relresfld 5196	Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
|- ( Rel R -> ( R |` U. U. R ) = R )
Theorem	relcoi2 5197	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
|- ( Rel R -> ( ( _I |` U. U. R ) o. R ) = R )
Theorem	relcoi1 5198	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
|- ( Rel R -> ( R o. ( _I |` U. U. R ) ) = R )
Theorem	unidmrn 5199	The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
|- U. U. `' A = ( dom A u. ran A )
Theorem	relcnvfld 5200	if R is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
|- ( Rel R -> U. U. R = U. U. `' R )