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	dfrn4 4801	Range defined in terms of image. (Contributed by NM, 14-May-2008.)
|- ran A = ( A " _V )
Theorem	csbrng 4802	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 4803	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 4804	The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
|- `' `' ( A |` B ) = ( `' `' A |` B )
Theorem	imacnvcnv 4805	The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
|- ( `' `' A " B ) = ( A " B )
Theorem	dmsnm 4806*	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 4807*	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 4808	The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
|- dom { (/) } = (/)
Theorem	cnvsn0 4809	The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- `' { (/) } = (/)
Theorem	dmsn0el 4810	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 4811*	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 4812	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 4813	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 4814	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 4815	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 4816	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 4817	Double converse of a singleton of an ordered pair. (Unlike cnvsn 4823, this does not need any sethood assumptions on A and B.) (Contributed by Mario Carneiro, 26-Apr-2015.)
|- `' `' { <. A , B >. } = `' { <. B , A >. }
Theorem	dmsnsnsng 4818	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 4819	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 4820	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 4821	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 4822	Extract the first member of an ordered pair. (See op2nda 4825 to extract the second member and op1stb 4227 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- U. dom { <. A , B >. } = A
Theorem	cnvsn 4823	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 4824	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4227 to extract the first member and op2nda 4825 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
|- A e. _V   &    |- B e. _V   =>    |- |^| |^| |^| `' { <. A , B >. } = B
Theorem	op2nda 4825	Extract the second member of an ordered pair. (See op1sta 4822 to extract the first member and op2ndb 4824 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 4826	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 4827	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 4828	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4829. (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 4829	Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4828 when the double intersection does not create class existence problems (caused by int0 3650). (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 4830	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 4831	A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
|- ( A |` dom A ) = `' `' A
Theorem	resdmres 4832	Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
|- ( A |` dom ( A |` B ) ) = ( A |` B )
Theorem	imadmres 4833	The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
|- ( A " dom ( A |` B ) ) = ( A " B )
Theorem	mptpreima 4834*	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 4835*	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 4836	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 4837*	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 4838*	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 4839	A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
|- Rel ( A o. B )
Theorem	dfco2 4840*	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 4841*	Generalization of dfco2 4840, 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 4842	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 4843	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 4844	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 4845	Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
|- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )
Theorem	imaco 4846	Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
|- ( ( A o. B ) " C ) = ( A " ( B " C ) )
Theorem	rnco 4847	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
|- ran ( A o. B ) = ran ( A |` ran B )
Theorem	rnco2 4848	The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
|- ran ( A o. B ) = ( A " ran B )
Theorem	dmco 4849	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 4850*	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 4851	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 4852	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 4853	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 4854	Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
|- ( A o. (/) ) = (/)
Theorem	co01 4855	Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
|- ( (/) o. A ) = (/)
Theorem	coi1 4856	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 4857	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 4858	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 4859	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 4860	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 4861	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 4862	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 4863	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	relrelss 4864	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 4865	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 4866	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 4867	Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
|- ( Rel R -> ( R |` U. U. R ) = R )
Theorem	relcoi2 4868	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 4869	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 4870	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 4871	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 )
Theorem	dfdm2 4872	Alternate definition of domain df-dm 4373 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
|- dom A = U. U. ( `' A o. A )
Theorem	unixpm 4873*	The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
|- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) )
Theorem	unixp0im 4874	The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
|- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) )
Theorem	cnvexg 4875	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
|- ( A e. V -> `' A e. _V )
Theorem	cnvex 4876	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
|- A e. _V   =>    |- `' A e. _V
Theorem	relcnvexb 4877	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
|- ( Rel R -> ( R e. _V <-> `' R e. _V ) )
Theorem	ressn 4878	Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( A |` { B } ) = ( { B } X. ( A " { B } ) )
Theorem	cnviinm 4879*	The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
|- ( E. y y e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )
Theorem	cnvpom 4880*	The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
|- ( E. x x e. A -> ( R Po A <-> `' R Po A ) )
Theorem	cnvsom 4881*	The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
|- ( E. x x e. A -> ( R Or A <-> `' R Or A ) )
Theorem	coexg 4882	The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
|- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V )
Theorem	coex 4883	The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( A o. B ) e. _V
Theorem	xpcom 4884*	Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
|- ( E. x x e. B -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) )
2.6.7  Definite description binder (inverted iota)
Syntax	cio 4885	Extend class notation with Russell's definition description binder (inverted iota).
class ( iota x ph )
Theorem	iotajust 4886*	Soundness justification theorem for df-iota 4887. (Contributed by Andrew Salmon, 29-Jun-2011.)
|- U. { y | { x | ph } = { y } } = U. { z | { x | ph } = { z } }
Definition	df-iota 4887*	Define Russell's definition description binder, which can be read as "the unique x such that ph," where ph ordinarily contains x as a free variable. Our definition is meaningful only when there is exactly one x such that ph is true (see iotaval 4898); otherwise, it evaluates to the empty set (see iotanul 4902). Russell used the inverted iota symbol iota to represent the binder. Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 4910 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( iota x ph ) = U. { y | { x | ph } = { y } }
Theorem	dfiota2 4888*	Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }
Theorem	nfiota1 4889	Bound-variable hypothesis builder for the iota class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x ( iota x ph )
Theorem	nfiotadxy 4890*	Deduction version of nfiotaxy 4891. (Contributed by Jim Kingdon, 21-Dec-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/_ x ( iota y ps ) )
Theorem	nfiotaxy 4891*	Bound-variable hypothesis builder for the iota class. (Contributed by NM, 23-Aug-2011.)
|- F/ x ph   =>    |- F/_ x ( iota y ph )
Theorem	cbviota 4892	Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( x = y -> ( ph <-> ps ) )   &    |- F/ y ph   &    |- F/ x ps   =>    |- ( iota x ph ) = ( iota y ps )
Theorem	cbviotav 4893*	Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( iota x ph ) = ( iota y ps )
Theorem	sb8iota 4894	Variable substitution in description binder. Compare sb8eu 1954. (Contributed by NM, 18-Mar-2013.)
|- F/ y ph   =>    |- ( iota x ph ) = ( iota y [ y / x ] ph )
Theorem	iotaeq 4895	Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )
Theorem	iotabi 4896	Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )
Theorem	uniabio 4897*	Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y )
Theorem	iotaval 4898*	Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
Theorem	iotauni 4899	Equivalence between two different forms of iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
Theorem	iotaint 4900	Equivalence between two different forms of iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )