P. 60 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ovmpoa 5901*	Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
|- ( ( x = A /\ y = B ) -> R = S )   &    |- F = ( x e. C , y e. D |-> R )   &    |- S e. _V   =>    |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )
Theorem	ovmpodf 5902*	Alternate deduction version of ovmpo 5906, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. C )   &    |- ( ( ph /\ x = A ) -> B e. D )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )   &    |- F/_ x F   &    |- F/ x ps   &    |- F/_ y F   &    |- F/ y ps   =>    |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )
Theorem	ovmpodv 5903*	Alternate deduction version of ovmpo 5906, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. C )   &    |- ( ( ph /\ x = A ) -> B e. D )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )   =>    |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )
Theorem	ovmpodv2 5904*	Alternate deduction version of ovmpo 5906, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. C )   &    |- ( ( ph /\ x = A ) -> B e. D )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )   =>    |- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ( A F B ) = S ) )
Theorem	ovmpog 5905*	Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
|- ( x = A -> R = G )   &    |- ( y = B -> G = S )   &    |- F = ( x e. C , y e. D |-> R )   =>    |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )
Theorem	ovmpo 5906*	Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
|- ( x = A -> R = G )   &    |- ( y = B -> G = S )   &    |- F = ( x e. C , y e. D |-> R )   &    |- S e. _V   =>    |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )
Theorem	ovi3 5907*	The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
|- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> S e. ( H X. H ) )   &    |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> R = S )   &    |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }   =>    |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( <. A , B >. F <. C , D >. ) = S )
Theorem	ov6g 5908*	The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)
|- ( <. x , y >. = <. A , B >. -> R = S )   &    |- F = { <. <. x , y >. , z >. | ( <. x , y >. e. C /\ z = R ) }   =>    |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( A F B ) = S )
Theorem	ovg 5909*	The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( ( ta /\ ( x e. R /\ y e. S ) ) -> E! z ph )   &    |- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }   =>    |- ( ( ta /\ ( A e. R /\ B e. S /\ C e. D ) ) -> ( ( A F B ) = C <-> th ) )
Theorem	ovres 5910	The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
|- ( ( A e. C /\ B e. D ) -> ( A ( F |` ( C X. D ) ) B ) = ( A F B ) )
Theorem	ovresd 5911	Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. X )   =>    |- ( ph -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
Theorem	oprssov 5912	The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)
|- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\ ( A e. C /\ B e. D ) ) -> ( A F B ) = ( A G B ) )
Theorem	fovrn 5913	An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)
|- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
Theorem	fovrnda 5914	An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
|- ( ph -> F : ( R X. S ) --> C )   =>    |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C )
Theorem	fovrnd 5915	An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
|- ( ph -> F : ( R X. S ) --> C )   &    |- ( ph -> A e. R )   &    |- ( ph -> B e. S )   =>    |- ( ph -> ( A F B ) e. C )
Theorem	fnrnov 5916*	The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)
|- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )
Theorem	foov 5917*	An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
|- ( F : ( A X. B ) -onto-> C <-> ( F : ( A X. B ) --> C /\ A. z e. C E. x e. A E. y e. B z = ( x F y ) ) )
Theorem	fnovrn 5918	An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)
|- ( ( F Fn ( A X. B ) /\ C e. A /\ D e. B ) -> ( C F D ) e. ran F )
Theorem	ovelrn 5919*	A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
|- ( F Fn ( A X. B ) -> ( C e. ran F <-> E. x e. A E. y e. B C = ( x F y ) ) )
Theorem	funimassov 5920*	Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)
|- ( ( Fun F /\ ( A X. B ) C_ dom F ) -> ( ( F " ( A X. B ) ) C_ C <-> A. x e. A A. y e. B ( x F y ) e. C ) )
Theorem	ovelimab 5921*	Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
|- ( ( F Fn A /\ ( B X. C ) C_ A ) -> ( D e. ( F " ( B X. C ) ) <-> E. x e. B E. y e. C D = ( x F y ) ) )
Theorem	ovconst2 5922	The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
|- C e. _V   =>    |- ( ( R e. A /\ S e. B ) -> ( R ( ( A X. B ) X. { C } ) S ) = C )
Theorem	caovclg 5923*	Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
|- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E )   =>    |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )
Theorem	caovcld 5924*	Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E )   &    |- ( ph -> A e. C )   &    |- ( ph -> B e. D )   =>    |- ( ph -> ( A F B ) e. E )
Theorem	caovcl 5925*	Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
|- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S )   =>    |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S )
Theorem	caovcomg 5926*	Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   =>    |- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )
Theorem	caovcomd 5927*	Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   =>    |- ( ph -> ( A F B ) = ( B F A ) )
Theorem	caovcom 5928*	Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
|- A e. _V   &    |- B e. _V   &    |- ( x F y ) = ( y F x )   =>    |- ( A F B ) = ( B F A )
Theorem	caovassg 5929*	Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   =>    |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
Theorem	caovassd 5930*	Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   =>    |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
Theorem	caovass 5931*	Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( ( x F y ) F z ) = ( x F ( y F z ) )   =>    |- ( ( A F B ) F C ) = ( A F ( B F C ) )
Theorem	caovcang 5932*	Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )   =>    |- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( A F B ) = ( A F C ) <-> B = C ) )
Theorem	caovcand 5933*	Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )   &    |- ( ph -> A e. T )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   =>    |- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) )
Theorem	caovcanrd 5934*	Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )   &    |- ( ph -> A e. T )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   =>    |- ( ph -> ( ( B F A ) = ( C F A ) <-> B = C ) )
Theorem	caovcan 5935*	Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
|- C e. _V   &    |- ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F z ) -> y = z ) )   =>    |- ( ( A e. S /\ B e. S ) -> ( ( A F B ) = ( A F C ) -> B = C ) )
Theorem	caovordig 5936*	Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y -> ( z F x ) R ( z F y ) ) )   =>    |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B -> ( C F A ) R ( C F B ) ) )
Theorem	caovordid 5937*	Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y -> ( z F x ) R ( z F y ) ) )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   =>    |- ( ph -> ( A R B -> ( C F A ) R ( C F B ) ) )
Theorem	caovordg 5938*	Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )   =>    |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B <-> ( C F A ) R ( C F B ) ) )
Theorem	caovordd 5939*	Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   =>    |- ( ph -> ( A R B <-> ( C F A ) R ( C F B ) ) )
Theorem	caovord2d 5940*	Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   =>    |- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) )
Theorem	caovord3d 5941*	Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ph -> D e. S )   =>    |- ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )
Theorem	caovord 5942*	Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
|- A e. _V   &    |- B e. _V   &    |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )   =>    |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )
Theorem	caovord2 5943*	Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
|- A e. _V   &    |- B e. _V   &    |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )   &    |- C e. _V   &    |- ( x F y ) = ( y F x )   =>    |- ( C e. S -> ( A R B <-> ( A F C ) R ( B F C ) ) )
Theorem	caovord3 5944*	Ordering law. (Contributed by NM, 29-Feb-1996.)
|- A e. _V   &    |- B e. _V   &    |- ( z e. S -> ( x R y <-> ( z F x ) R ( z F y ) ) )   &    |- C e. _V   &    |- ( x F y ) = ( y F x )   &    |- D e. _V   =>    |- ( ( ( B e. S /\ C e. S ) /\ ( A F B ) = ( C F D ) ) -> ( A R C <-> D R B ) )
Theorem	caovdig 5945*	Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
|- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) )   =>    |- ( ( ph /\ ( A e. K /\ B e. S /\ C e. S ) ) -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )
Theorem	caovdid 5946*	Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) )   &    |- ( ph -> A e. K )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   =>    |- ( ph -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )
Theorem	caovdir2d 5947*	Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) = ( y G x ) )   =>    |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) )
Theorem	caovdirg 5948*	Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) -> ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) )   =>    |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. K ) ) -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) )
Theorem	caovdird 5949*	Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) -> ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. K )   =>    |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) )
Theorem	caovdi 5950*	Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) )   =>    |- ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) )
Theorem	caov32d 5951*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   =>    |- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )
Theorem	caov12d 5952*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   =>    |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )
Theorem	caov31d 5953*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   =>    |- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) )
Theorem	caov13d 5954*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   =>    |- ( ph -> ( A F ( B F C ) ) = ( C F ( B F A ) ) )
Theorem	caov4d 5955*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   &    |- ( ph -> D e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )   =>    |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )
Theorem	caov411d 5956*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   &    |- ( ph -> D e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )   =>    |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) )
Theorem	caov42d 5957*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   &    |- ( ph -> D e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )   =>    |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) ) )
Theorem	caov32 5958*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x F y ) = ( y F x )   &    |- ( ( x F y ) F z ) = ( x F ( y F z ) )   =>    |- ( ( A F B ) F C ) = ( ( A F C ) F B )
Theorem	caov12 5959*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x F y ) = ( y F x )   &    |- ( ( x F y ) F z ) = ( x F ( y F z ) )   =>    |- ( A F ( B F C ) ) = ( B F ( A F C ) )
Theorem	caov31 5960*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x F y ) = ( y F x )   &    |- ( ( x F y ) F z ) = ( x F ( y F z ) )   =>    |- ( ( A F B ) F C ) = ( ( C F B ) F A )
Theorem	caov13 5961*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x F y ) = ( y F x )   &    |- ( ( x F y ) F z ) = ( x F ( y F z ) )   =>    |- ( A F ( B F C ) ) = ( C F ( B F A ) )
Theorem	caovdilemd 5962*	Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) = ( y G x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) G z ) = ( ( x G z ) F ( y G z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x G y ) G z ) = ( x G ( y G z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) e. S )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ph -> D e. S )   &    |- ( ph -> H e. S )   =>    |- ( ph -> ( ( ( A G C ) F ( B G D ) ) G H ) = ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) )
Theorem	caovlem2d 5963*	Rearrangement of expression involving multiplication ( G) and addition ( F). (Contributed by Jim Kingdon, 3-Jan-2020.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) = ( y G x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) G z ) = ( ( x G z ) F ( y G z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x G y ) G z ) = ( x G ( y G z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) e. S )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ph -> D e. S )   &    |- ( ph -> H e. S )   &    |- ( ph -> R e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )   =>    |- ( ph -> ( ( ( ( A G C ) F ( B G D ) ) G H ) F ( ( ( A G D ) F ( B G C ) ) G R ) ) = ( ( A G ( ( C G H ) F ( D G R ) ) ) F ( B G ( ( C G R ) F ( D G H ) ) ) ) )
Theorem	caovimo 5964*	Uniqueness of inverse element in commutative, associative operation with identity. The identity element is B. (Contributed by Jim Kingdon, 18-Sep-2019.)
|- B e. S   &    |- ( ( x e. S /\ y e. S ) -> ( x F y ) = ( y F x ) )   &    |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   &    |- ( x e. S -> ( x F B ) = x )   =>    |- ( A e. S -> E* w ( w e. S /\ ( A F w ) = B ) )
Theorem	grprinvlem 5965*	Lemma for grprinvd 5966. (Contributed by NM, 9-Aug-2013.)
|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ph -> O e. B )   &    |- ( ( ph /\ x e. B ) -> ( O .+ x ) = x )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O )   &    |- ( ( ph /\ ps ) -> X e. B )   &    |- ( ( ph /\ ps ) -> ( X .+ X ) = X )   =>    |- ( ( ph /\ ps ) -> X = O )
Theorem	grprinvd 5966*	Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ph -> O e. B )   &    |- ( ( ph /\ x e. B ) -> ( O .+ x ) = x )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O )   &    |- ( ( ph /\ ps ) -> X e. B )   &    |- ( ( ph /\ ps ) -> N e. B )   &    |- ( ( ph /\ ps ) -> ( N .+ X ) = O )   =>    |- ( ( ph /\ ps ) -> ( X .+ N ) = O )
Theorem	grpridd 5967*	Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ph -> O e. B )   &    |- ( ( ph /\ x e. B ) -> ( O .+ x ) = x )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O )   =>    |- ( ( ph /\ x e. B ) -> ( x .+ O ) = x )
2.6.11  Maps-to notation
Theorem	elmpocl 5968*	If a two-parameter class is inhabited, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( X e. ( S F T ) -> ( S e. A /\ T e. B ) )
Theorem	elmpocl1 5969*	If a two-parameter class is inhabited, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( X e. ( S F T ) -> S e. A )
Theorem	elmpocl2 5970*	If a two-parameter class is inhabited, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( X e. ( S F T ) -> T e. B )
Theorem	elovmpo 5971*	Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.)
|- D = ( a e. A , b e. B |-> C )   &    |- C e. _V   &    |- ( ( a = X /\ b = Y ) -> C = E )   =>    |- ( F e. ( X D Y ) <-> ( X e. A /\ Y e. B /\ F e. E ) )
Theorem	f1ocnvd 5972*	Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- F = ( x e. A |-> C )   &    |- ( ( ph /\ x e. A ) -> C e. W )   &    |- ( ( ph /\ y e. B ) -> D e. X )   &    |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )   =>    |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
Theorem	f1od 5973*	Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
|- F = ( x e. A |-> C )   &    |- ( ( ph /\ x e. A ) -> C e. W )   &    |- ( ( ph /\ y e. B ) -> D e. X )   &    |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )   =>    |- ( ph -> F : A -1-1-onto-> B )
Theorem	f1ocnv2d 5974*	Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- F = ( x e. A |-> C )   &    |- ( ( ph /\ x e. A ) -> C e. B )   &    |- ( ( ph /\ y e. B ) -> D e. A )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )   =>    |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
Theorem	f1o2d 5975*	Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
|- F = ( x e. A |-> C )   &    |- ( ( ph /\ x e. A ) -> C e. B )   &    |- ( ( ph /\ y e. B ) -> D e. A )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )   =>    |- ( ph -> F : A -1-1-onto-> B )
Theorem	f1opw2 5976*	A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5977 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
|- ( ph -> F : A -1-1-onto-> B )   &    |- ( ph -> ( `' F " a ) e. _V )   &    |- ( ph -> ( F " b ) e. _V )   =>    |- ( ph -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )
Theorem	f1opw 5977*	A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- ( F : A -1-1-onto-> B -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )
Theorem	suppssfv 5978*	Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )   &    |- ( ph -> ( F ` Y ) = Z )   &    |- ( ( ph /\ x e. D ) -> A e. V )   =>    |- ( ph -> ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) C_ L )
Theorem	suppssov1 5979*	Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )   &    |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )   &    |- ( ( ph /\ x e. D ) -> A e. V )   &    |- ( ( ph /\ x e. D ) -> B e. R )   =>    |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ L )
2.6.12  Function operation
Syntax	cof 5980	Extend class notation to include mapping of an operation to a function operation.
class oF R
Syntax	cofr 5981	Extend class notation to include mapping of a binary relation to a function relation.
class oR R
Definition	df-of 5982*	Define the function operation map. The definition is designed so that if R is a binary operation, then oF R is the analogous operation on functions which corresponds to applying R pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- oF R = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
Definition	df-ofr 5983*	Define the function relation map. The definition is designed so that if R is a binary relation, then oF R is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
Theorem	ofeq 5984	Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( R = S -> oF R = oF S )
Theorem	ofreq 5985	Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( R = S -> oR R = oR S )
Theorem	ofexg 5986	A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
|- ( A e. V -> ( oF R |` A ) e. _V )
Theorem	nfof 5987	Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- F/_ x R   =>    |- F/_ x oF R
Theorem	nfofr 5988	Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- F/_ x R   =>    |- F/_ x oR R
Theorem	offval 5989*	Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )   &    |- ( ( ph /\ x e. B ) -> ( G ` x ) = D )   =>    |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) )
Theorem	ofrfval 5990*	Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )   &    |- ( ( ph /\ x e. B ) -> ( G ` x ) = D )   =>    |- ( ph -> ( F oR R G <-> A. x e. S C R D ) )
Theorem	ofvalg 5991	Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )   &    |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )   &    |- ( ( ph /\ X e. S ) -> ( C R D ) e. U )   =>    |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )
Theorem	ofrval 5992	Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )   &    |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )   =>    |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D )
Theorem	ofmresval 5993	Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
|- ( ph -> F e. A )   &    |- ( ph -> G e. B )   =>    |- ( ph -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) )
Theorem	off 5994*	The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : B --> T )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = C   =>    |- ( ph -> ( F oF R G ) : C --> U )
Theorem	offeq 5995*	Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
|- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : B --> T )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = C   &    |- ( ph -> H : C --> U )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = D )   &    |- ( ( ph /\ x e. B ) -> ( G ` x ) = E )   &    |- ( ( ph /\ x e. C ) -> ( D R E ) = ( H ` x ) )   =>    |- ( ph -> ( F oF R G ) = H )
Theorem	ofres 5996	Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = C   =>    |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) )
Theorem	offval2 5997*	The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x e. A ) -> B e. W )   &    |- ( ( ph /\ x e. A ) -> C e. X )   &    |- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ph -> G = ( x e. A |-> C ) )   =>    |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )
Theorem	ofrfval2 5998*	The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x e. A ) -> B e. W )   &    |- ( ( ph /\ x e. A ) -> C e. X )   &    |- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ph -> G = ( x e. A |-> C ) )   =>    |- ( ph -> ( F oR R G <-> A. x e. A B R C ) )
Theorem	suppssof1 5999*	Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- ( ph -> ( `' A " ( _V \ { Y } ) ) C_ L )   &    |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )   &    |- ( ph -> A : D --> V )   &    |- ( ph -> B : D --> R )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L )
Theorem	ofco 6000	The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> H : D --> C )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> D e. X )   &    |- ( A i^i B ) = C   =>    |- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )