P. 58 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5701-5800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	oveq2i 5701	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( C F A ) = ( C F B )
Theorem	oveq12i 5702	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- A = B   &    |- C = D   =>    |- ( A F C ) = ( B F D )
Theorem	oveqi 5703	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
|- A = B   =>    |- ( C A D ) = ( C B D )
Theorem	oveq123i 5704	Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
|- A = C   &    |- B = D   &    |- F = G   =>    |- ( A F B ) = ( C G D )
Theorem	oveq1d 5705	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A F C ) = ( B F C ) )
Theorem	oveq2d 5706	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C F A ) = ( C F B ) )
Theorem	oveqd 5707	Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D ) = ( C B D ) )
Theorem	oveq12d 5708	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A F C ) = ( B F D ) )
Theorem	oveqan12d 5709	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A F C ) = ( B F D ) )
Theorem	oveqan12rd 5710	Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ps /\ ph ) -> ( A F C ) = ( B F D ) )
Theorem	oveq123d 5711	Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A F C ) = ( B G D ) )
Theorem	fvoveq1d 5712	Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )
Theorem	fvoveq1 5713	Equality theorem for nested function and operation value. Closed form of fvoveq1d 5712. (Contributed by AV, 23-Jul-2022.)
|- ( A = B -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )
Theorem	ovanraleqv 5714*	Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
|- ( B = X -> ( ph <-> ps ) )   =>    |- ( B = X -> ( A. x e. V ( ph /\ ( A .x. B ) = C ) <-> A. x e. V ( ps /\ ( A .x. X ) = C ) ) )
Theorem	imbrov2fvoveq 5715	Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
|- ( X = Y -> ( ph <-> ps ) )   =>    |- ( X = Y -> ( ( ph -> ( F ` ( ( G ` X ) .x. O ) ) R A ) <-> ( ps -> ( F ` ( ( G ` Y ) .x. O ) ) R A ) ) )
Theorem	nfovd 5716	Deduction version of bound-variable hypothesis builder nfov 5717. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x F )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x ( A F B ) )
Theorem	nfov 5717	Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
|- F/_ x A   &    |- F/_ x F   &    |- F/_ x B   =>    |- F/_ x ( A F B )
Theorem	oprabidlem 5718*	Slight elaboration of exdistrfor 1735. A lemma for oprabid 5719. (Contributed by Jim Kingdon, 15-Jan-2019.)
|- ( E. x E. y ( x = z /\ ps ) -> E. x ( x = z /\ E. y ps ) )
Theorem	oprabid 5719	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between x, y, and z, we use ax-bndl 1451 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
|- ( <. <. x , y >. , z >. e. { <. <. x , y >. , z >. | ph } <-> ph )
Theorem	fnovex 5720	The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
|- ( ( F Fn ( C X. D ) /\ A e. C /\ B e. D ) -> ( A F B ) e. _V )
Theorem	ovexg 5721	Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
|- ( ( A e. V /\ F e. W /\ B e. X ) -> ( A F B ) e. _V )
Theorem	ovprc 5722	The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- Rel dom F   =>    |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) )
Theorem	ovprc1 5723	The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
|- Rel dom F   =>    |- ( -. A e. _V -> ( A F B ) = (/) )
Theorem	ovprc2 5724	The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- Rel dom F   =>    |- ( -. B e. _V -> ( A F B ) = (/) )
Theorem	csbov123g 5725	Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
|- ( A e. D -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) )
Theorem	csbov12g 5726*	Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
|- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F [_ A / x ]_ C ) )
Theorem	csbov1g 5727*	Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
|- ( A e. V -> [_ A / x ]_ ( B F C ) = ( [_ A / x ]_ B F C ) )
Theorem	csbov2g 5728*	Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
|- ( A e. V -> [_ A / x ]_ ( B F C ) = ( B F [_ A / x ]_ C ) )
Theorem	rspceov 5729*	A frequently used special case of rspc2ev 2750 for operation values. (Contributed by NM, 21-Mar-2007.)
|- ( ( C e. A /\ D e. B /\ S = ( C F D ) ) -> E. x e. A E. y e. B S = ( x F y ) )
Theorem	fnotovb 5730	Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5381. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( ( C F D ) = R <-> <. C , D , R >. e. F ) )
Theorem	opabbrex 5731*	A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
|- ( ( V e. _V /\ E e. _V ) -> ( f ( V W E ) p -> th ) )   &    |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | th } e. _V )   =>    |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V )
Theorem	0neqopab 5732	The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
|- -. (/) e. { <. x , y >. | ph }
Theorem	brabvv 5733*	If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.)
|- ( X { <. x , y >. | ph } Y -> ( X e. _V /\ Y e. _V ) )
Theorem	dfoprab2 5734*	Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)
|- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
Theorem	reloprab 5735*	An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
|- Rel { <. <. x , y >. , z >. | ph }
Theorem	nfoprab1 5736	The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
|- F/_ x { <. <. x , y >. , z >. | ph }
Theorem	nfoprab2 5737	The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)
|- F/_ y { <. <. x , y >. , z >. | ph }
Theorem	nfoprab3 5738	The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
|- F/_ z { <. <. x , y >. , z >. | ph }
Theorem	nfoprab 5739*	Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
|- F/ w ph   =>    |- F/_ w { <. <. x , y >. , z >. | ph }
Theorem	oprabbid 5740*	Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)
|- F/ x ph   &    |- F/ y ph   &    |- F/ z ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )
Theorem	oprabbidv 5741*	Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )
Theorem	oprabbii 5742*	Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
|- ( ph <-> ps )   =>    |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , z >. | ps }
Theorem	ssoprab2 5743	Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4126. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
|- ( A. x A. y A. z ( ph -> ps ) -> { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps } )
Theorem	ssoprab2b 5744	Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4127. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
|- ( { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph -> ps ) )
Theorem	eqoprab2b 5745	Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4130. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph <-> ps ) )
Theorem	mpt2eq123 5746*	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)
|- ( ( A = D /\ A. x e. A ( B = E /\ A. y e. B C = F ) ) -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )
Theorem	mpt2eq12 5747*	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
|- ( ( A = C /\ B = D ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )
Theorem	mpt2eq123dva 5748*	An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
|- ( ph -> A = D )   &    |- ( ( ph /\ x e. A ) -> B = E )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F )   =>    |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )
Theorem	mpt2eq123dv 5749*	An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.)
|- ( ph -> A = D )   &    |- ( ph -> B = E )   &    |- ( ph -> C = F )   =>    |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )
Theorem	mpt2eq123i 5750	An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.)
|- A = D   &    |- B = E   &    |- C = F   =>    |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F )
Theorem	mpt2eq3dva 5751*	Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.)
|- ( ( ph /\ x e. A /\ y e. B ) -> C = D )   =>    |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )
Theorem	mpt2eq3ia 5752	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
|- ( ( x e. A /\ y e. B ) -> C = D )   =>    |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )
Theorem	nfmpt21 5753	Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
|- F/_ x ( x e. A , y e. B |-> C )
Theorem	nfmpt22 5754	Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
|- F/_ y ( x e. A , y e. B |-> C )
Theorem	nfmpt2 5755*	Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
|- F/_ z A   &    |- F/_ z B   &    |- F/_ z C   =>    |- F/_ z ( x e. A , y e. B |-> C )
Theorem	mpt20 5756	A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
|- ( x e. (/) , y e. B |-> C ) = (/)
Theorem	oprab4 5757*	Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)
|- { <. <. x , y >. , z >. | ( <. x , y >. e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }
Theorem	cbvoprab1 5758*	Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
|- F/ w ph   &    |- F/ x ps   &    |- ( x = w -> ( ph <-> ps ) )   =>    |- { <. <. x , y >. , z >. | ph } = { <. <. w , y >. , z >. | ps }
Theorem	cbvoprab2 5759*	Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- F/ w ph   &    |- F/ y ps   &    |- ( y = w -> ( ph <-> ps ) )   =>    |- { <. <. x , y >. , z >. | ph } = { <. <. x , w >. , z >. | ps }
Theorem	cbvoprab12 5760*	Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- F/ w ph   &    |- F/ v ph   &    |- F/ x ps   &    |- F/ y ps   &    |- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )   =>    |- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }
Theorem	cbvoprab12v 5761*	Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)
|- ( ( x = w /\ y = v ) -> ( ph <-> ps ) )   =>    |- { <. <. x , y >. , z >. | ph } = { <. <. w , v >. , z >. | ps }
Theorem	cbvoprab3 5762*	Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
|- F/ w ph   &    |- F/ z ps   &    |- ( z = w -> ( ph <-> ps ) )   =>    |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }
Theorem	cbvoprab3v 5763*	Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)
|- ( z = w -> ( ph <-> ps ) )   =>    |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }
Theorem	cbvmpt2x 5764*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5765 allows B to be a function of x. (Contributed by NM, 29-Dec-2014.)
|- F/_ z B   &    |- F/_ x D   &    |- F/_ z C   &    |- F/_ w C   &    |- F/_ x E   &    |- F/_ y E   &    |- ( x = z -> B = D )   &    |- ( ( x = z /\ y = w ) -> C = E )   =>    |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. D |-> E )
Theorem	cbvmpt2 5765*	Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
|- F/_ z C   &    |- F/_ w C   &    |- F/_ x D   &    |- F/_ y D   &    |- ( ( x = z /\ y = w ) -> C = D )   =>    |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )
Theorem	cbvmpt2v 5766*	Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3955, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
|- ( x = z -> C = E )   &    |- ( y = w -> E = D )   =>    |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )
Theorem	dmoprab 5767*	The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
|- dom { <. <. x , y >. , z >. | ph } = { <. x , y >. | E. z ph }
Theorem	dmoprabss 5768*	The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
|- dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } C_ ( A X. B )
Theorem	rnoprab 5769*	The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
|- ran { <. <. x , y >. , z >. | ph } = { z | E. x E. y ph }
Theorem	rnoprab2 5770*	The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
|- ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { z | E. x e. A E. y e. B ph }
Theorem	reldmoprab 5771*	The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
|- Rel dom { <. <. x , y >. , z >. | ph }
Theorem	oprabss 5772*	Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
|- { <. <. x , y >. , z >. | ph } C_ ( ( _V X. _V ) X. _V )
Theorem	eloprabga 5773*	The law of concretion for operation class abstraction. Compare elopab 4109. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )
Theorem	eloprabg 5774*	The law of concretion for operation class abstraction. Compare elopab 4109. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) )
Theorem	ssoprab2i 5775*	Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)
|- ( ph -> ps )   =>    |- { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps }
Theorem	mpt2v 5776*	Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
|- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | z = C }
Theorem	mpt2mptx 5777*	Express a two-argument function as a one-argument function, or vice-versa. In this version B ( x ) is not assumed to be constant w.r.t x. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- ( z = <. x , y >. -> C = D )   =>    |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )
Theorem	mpt2mpt 5778*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
|- ( z = <. x , y >. -> C = D )   =>    |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D )
Theorem	resoprab 5779*	Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
|- ( { <. <. x , y >. , z >. | ph } |` ( A X. B ) ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }
Theorem	resoprab2 5780*	Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( C C_ A /\ D C_ B ) -> ( { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } |` ( C X. D ) ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ ph ) } )
Theorem	resmpt2 5781*	Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
|- ( ( C C_ A /\ D C_ B ) -> ( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) = ( x e. C , y e. D |-> E ) )
Theorem	funoprabg 5782*	"At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
|- ( A. x A. y E* z ph -> Fun { <. <. x , y >. , z >. | ph } )
Theorem	funoprab 5783*	"At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
|- E* z ph   =>    |- Fun { <. <. x , y >. , z >. | ph }
Theorem	fnoprabg 5784*	Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)
|- ( A. x A. y ( ph -> E! z ps ) -> { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn { <. x , y >. | ph } )
Theorem	mpt2fun 5785*	The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
|- F = ( x e. A , y e. B |-> C )   =>    |- Fun F
Theorem	fnoprab 5786*	Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.)
|- ( ph -> E! z ps )   =>    |- { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn { <. x , y >. | ph }
Theorem	ffnov 5787*	An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)
|- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) e. C ) )
Theorem	fovcl 5788	Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
|- F : ( R X. S ) --> C   =>    |- ( ( A e. R /\ B e. S ) -> ( A F B ) e. C )
Theorem	eqfnov 5789*	Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
|- ( ( F Fn ( A X. B ) /\ G Fn ( C X. D ) ) -> ( F = G <-> ( ( A X. B ) = ( C X. D ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) )
Theorem	eqfnov2 5790*	Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)
|- ( ( F Fn ( A X. B ) /\ G Fn ( A X. B ) ) -> ( F = G <-> A. x e. A A. y e. B ( x F y ) = ( x G y ) ) )
Theorem	fnovim 5791*	Representation of a function in terms of its values. (Contributed by Jim Kingdon, 16-Jan-2019.)
|- ( F Fn ( A X. B ) -> F = ( x e. A , y e. B |-> ( x F y ) ) )
Theorem	mpt22eqb 5792*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5790. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( A. x e. A A. y e. B C e. V -> ( ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) <-> A. x e. A A. y e. B C = D ) )
Theorem	rnmpt2 5793*	The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ran F = { z | E. x e. A E. y e. B z = C }
Theorem	reldmmpt2 5794*	The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- F = ( x e. A , y e. B |-> C )   =>    |- Rel dom F
Theorem	elrnmpt2g 5795*	Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( D e. V -> ( D e. ran F <-> E. x e. A E. y e. B D = C ) )
Theorem	elrnmpt2 5796*	Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. A , y e. B |-> C )   &    |- C e. _V   =>    |- ( D e. ran F <-> E. x e. A E. y e. B D = C )
Theorem	ralrnmpt2 5797*	A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
|- F = ( x e. A , y e. B |-> C )   &    |- ( z = C -> ( ph <-> ps ) )   =>    |- ( A. x e. A A. y e. B C e. V -> ( A. z e. ran F ph <-> A. x e. A A. y e. B ps ) )
Theorem	rexrnmpt2 5798*	A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.)
|- F = ( x e. A , y e. B |-> C )   &    |- ( z = C -> ( ph <-> ps ) )   =>    |- ( A. x e. A A. y e. B C e. V -> ( E. z e. ran F ph <-> E. x e. A E. y e. B ps ) )
Theorem	ovid 5799*	The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
|- ( ( x e. R /\ y e. S ) -> E! z ph )   &    |- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }   =>    |- ( ( x e. R /\ y e. S ) -> ( ( x F y ) = z <-> ph ) )
Theorem	ovidig 5800*	The value of an operation class abstraction. Compare ovidi 5801. The condition ( x e. R /\ y e. S ) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- E* z ph   &    |- F = { <. <. x , y >. , z >. | ph }   =>    |- ( ph -> ( x F y ) = z )