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	rnsnop 4801	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 4802	Extract the first member of an ordered pair. (See op2nda 4805 to extract the second member and op1stb 4209 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- U. dom { <. A , B >. } = A
Theorem	cnvsn 4803	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 4804	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4209 to extract the first member and op2nda 4805 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
|- A e. _V   &    |- B e. _V   =>    |- |^| |^| |^| `' { <. A , B >. } = B
Theorem	op2nda 4805	Extract the second member of an ordered pair. (See op1sta 4802 to extract the first member and op2ndb 4804 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 4806	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 4807	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 4808	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4809. (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 4809	Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4808 when the double intersection does not create class existence problems (caused by int0 3629). (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 4810	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 4811	A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
|- ( A |` dom A ) = `' `' A
Theorem	resdmres 4812	Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
|- ( A |` dom ( A |` B ) ) = ( A |` B )
Theorem	imadmres 4813	The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
|- ( A " dom ( A |` B ) ) = ( A " B )
Theorem	mptpreima 4814*	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 4815*	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 4816	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 4817*	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 4818*	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 4819	A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
|- Rel ( A o. B )
Theorem	dfco2 4820*	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 4821*	Generalization of dfco2 4820, 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 4822	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 4823	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 4824	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 4825	Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
|- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )
Theorem	imaco 4826	Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
|- ( ( A o. B ) " C ) = ( A " ( B " C ) )
Theorem	rnco 4827	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
|- ran ( A o. B ) = ran ( A |` ran B )
Theorem	rnco2 4828	The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
|- ran ( A o. B ) = ( A " ran B )
Theorem	dmco 4829	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 4830*	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 4831	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 4832	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 4833	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 4834	Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
|- ( A o. (/) ) = (/)
Theorem	co01 4835	Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
|- ( (/) o. A ) = (/)
Theorem	coi1 4836	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 4837	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 4838	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 4839	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 4840	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 4841	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 4842	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 4843	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 4844	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 4845	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 4846	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 4847	Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
|- ( Rel R -> ( R |` U. U. R ) = R )
Theorem	relcoi2 4848	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 4849	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 4850	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 4851	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 4852	Alternate definition of domain df-dm 4355 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
|- dom A = U. U. ( `' A o. A )
Theorem	unixpm 4853*	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 4854	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 4855	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 4856	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 4857	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 4858	Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( A |` { B } ) = ( { B } X. ( A " { B } ) )
Theorem	cnviinm 4859*	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 4860*	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 4861*	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 4862	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 4863	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 4864*	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 4865	Extend class notation with Russell's definition description binder (inverted iota).
class ( iota x ph )
Theorem	iotajust 4866*	Soundness justification theorem for df-iota 4867. (Contributed by Andrew Salmon, 29-Jun-2011.)
|- U. { y | { x | ph } = { y } } = U. { z | { x | ph } = { z } }
Definition	df-iota 4867*	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 4878); otherwise, it evaluates to the empty set (see iotanul 4882). 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 4890 (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 4868*	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 4869	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 4870*	Deduction version of nfiotaxy 4871. (Contributed by Jim Kingdon, 21-Dec-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/_ x ( iota y ps ) )
Theorem	nfiotaxy 4871*	Bound-variable hypothesis builder for the iota class. (Contributed by NM, 23-Aug-2011.)
|- F/ x ph   =>    |- F/_ x ( iota y ph )
Theorem	cbviota 4872	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 4873*	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 4874	Variable substitution in description binder. Compare sb8eu 1913. (Contributed by NM, 18-Mar-2013.)
|- F/ y ph   =>    |- ( iota x ph ) = ( iota y [ y / x ] ph )
Theorem	iotaeq 4875	Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )
Theorem	iotabi 4876	Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )
Theorem	uniabio 4877*	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 4878*	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 4879	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 4880	Equivalence between two different forms of iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )
Theorem	iota1 4881	Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
Theorem	iotanul 4882	Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one x that satisfies ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( -. E! x ph -> ( iota x ph ) = (/) )
Theorem	euiotaex 4883	Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the iota class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
|- ( E! x ph -> ( iota x ph ) e. _V )
Theorem	iotass 4884*	Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
|- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )
Theorem	iota4 4885	Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
Theorem	iota4an 4886	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )
Theorem	iota5 4887*	A method for computing iota. (Contributed by NM, 17-Sep-2013.)
|- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) )   =>    |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A )
Theorem	iotabidv 4888*	Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( iota x ps ) = ( iota x ch ) )
Theorem	iotabii 4889	Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ph <-> ps )   =>    |- ( iota x ph ) = ( iota x ps )
Theorem	iotacl 4890	Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 4867). (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( E! x ph -> ( iota x ph ) e. { x | ph } )
Theorem	iota2df 4891	A condition that allows us to represent "the unique element such that ph " with a class expression A. (Contributed by NM, 30-Dec-2014.)
|- ( ph -> B e. V )   &    |- ( ph -> E! x ps )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   &    |- F/ x ph   &    |- ( ph -> F/ x ch )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> ( ch <-> ( iota x ps ) = B ) )
Theorem	iota2d 4892*	A condition that allows us to represent "the unique element such that ph " with a class expression A. (Contributed by NM, 30-Dec-2014.)
|- ( ph -> B e. V )   &    |- ( ph -> E! x ps )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( ch <-> ( iota x ps ) = B ) )
Theorem	iota2 4893*	The unique element such that ph. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
Theorem	sniota 4894	A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- ( E! x ph -> { x | ph } = { ( iota x ph ) } )
Theorem	csbiotag 4895*	Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
|- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )
2.6.8  Functions
Syntax	wfun 4896	Extend the definition of a wff to include the function predicate. (Read: A is a function.)
wff Fun A
Syntax	wfn 4897	Extend the definition of a wff to include the function predicate with a domain. (Read: A is a function on B.)
wff A Fn B
Syntax	wf 4898	Extend the definition of a wff to include the function predicate with domain and codomain. (Read: F maps A into B.)
wff F : A --> B
Syntax	wf1 4899	Extend the definition of a wff to include one-to-one functions. (Read: F maps A one-to-one into B.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
wff F : A -1-1-> B
Syntax	wfo 4900	Extend the definition of a wff to include onto functions. (Read: F maps A onto B.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
wff F : A -onto-> B