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Theorem List for Metamath Proof Explorer - 5901-6000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	resoprab 5901*	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 5902*	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 5903*	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 5904*	"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 5905*	"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 5906*	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 5907*	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 5908*	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 5909*	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 5910	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 5911*	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 5912*	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	fnov 5913*	Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( F Fn ( A X. B ) <-> F = ( x e. A , y e. B |-> ( x F y ) ) )
Theorem	mpt22eqb 5914*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5912. (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 5915*	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 5916*	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 5917*	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 5918*	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 5919*	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 5920*	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	oprabexd 5921*	Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> A e. _V )   &    |- ( ph -> B e. _V )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> E* z ps )   &    |- ( ph -> F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } )   =>    |- ( ph -> F e. _V )
Theorem	oprabex 5922*	Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x e. A /\ y e. B ) -> E* z ph )   &    |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }   =>    |- F e. _V
Theorem	oprabex3 5923*	Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
|- H e. _V   &    |- 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 ) ) }   =>    |- F e. _V
Theorem	oprabrexex2 5924*	Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
|- A e. _V   &    |- { <. <. x , y >. , z >. | ph } e. _V   =>    |- { <. <. x , y >. , z >. | E. w e. A ph } e. _V
Theorem	ovid 5925*	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 5926*	The value of an operation class abstraction. Compare ovidi 5927. 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 )
Theorem	ovidi 5927*	The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)
|- ( ( 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 ) -> ( ph -> ( x F y ) = z ) )
Theorem	ov 5928*	The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
|- C e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> th ) )   &    |- ( ( x e. R /\ y e. S ) -> E! z ph )   &    |- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }   =>    |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) )
Theorem	ovigg 5929*	The value of an operation class abstraction. Compare ovig 5930. The condition ( x e. R /\ y e. S ) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   &    |- E* z ph   &    |- F = { <. <. x , y >. , z >. | ph }   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> ( A F B ) = C ) )
Theorem	ovig 5930*	The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   &    |- ( ( x e. R /\ y e. S ) -> E* z ph )   &    |- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }   =>    |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ps -> ( A F B ) = C ) )
Theorem	ovmpt4g 5931*	Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5569.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )
Theorem	ovmpt2s 5932*	Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- F = ( x e. C , y e. D |-> R )   =>    |- ( ( A e. C /\ B e. D /\ [_ A / x ]_ [_ B / y ]_ R e. V ) -> ( A F B ) = [_ A / x ]_ [_ B / y ]_ R )
Theorem	ov2gf 5933*	The value of an operation class abstraction. A version of ovmpt2g 5943 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ y B   &    |- F/_ x G   &    |- F/_ y S   &    |- ( 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	ovmpt2dxf 5934*	Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- ( ph -> F = ( x e. C , y e. D |-> R ) )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )   &    |- ( ( ph /\ x = A ) -> D = L )   &    |- ( ph -> A e. C )   &    |- ( ph -> B e. L )   &    |- ( ph -> S e. X )   &    |- F/ x ph   &    |- F/ y ph   &    |- F/_ y A   &    |- F/_ x B   &    |- F/_ x S   &    |- F/_ y S   =>    |- ( ph -> ( A F B ) = S )
Theorem	ovmpt2dx 5935*	Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- ( ph -> F = ( x e. C , y e. D |-> R ) )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )   &    |- ( ( ph /\ x = A ) -> D = L )   &    |- ( ph -> A e. C )   &    |- ( ph -> B e. L )   &    |- ( ph -> S e. X )   =>    |- ( ph -> ( A F B ) = S )
Theorem	ovmpt2d 5936*	Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- ( ph -> F = ( x e. C , y e. D |-> R ) )   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )   &    |- ( ph -> A e. C )   &    |- ( ph -> B e. D )   &    |- ( ph -> S e. X )   =>    |- ( ph -> ( A F B ) = S )
Theorem	ovmpt2x 5937*	The value of an operation class abstraction. Variant of ovmpt2ga 5938 which does not require D and x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
|- ( ( x = A /\ y = B ) -> R = S )   &    |- ( x = A -> D = L )   &    |- F = ( x e. C , y e. D |-> R )   =>    |- ( ( A e. C /\ B e. L /\ S e. H ) -> ( A F B ) = S )
Theorem	ovmpt2ga 5938*	Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)
|- ( ( x = A /\ y = B ) -> R = S )   &    |- F = ( x e. C , y e. D |-> R )   =>    |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )
Theorem	ovmpt2a 5939*	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	ovmpt2df 5940*	Alternate deduction version of ovmpt2 5944, 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	ovmpt2dv 5941*	Alternate deduction version of ovmpt2 5944, 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	ovmpt2dv2 5942*	Alternate deduction version of ovmpt2 5944, 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	ovmpt2g 5943*	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	ovmpt2 5944*	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	ov3 5945*	The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
|- S e. _V   &    |- ( ( ( 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 5946*	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 5947*	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 5948	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 5949	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 5950	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 5951	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 5952	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 5953	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 5954*	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 5955*	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 5956	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 5957*	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 5958*	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 5959*	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 5960	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	ab2rexex 5961*	Existence of a class abstraction of existentially restricted sets. Variables x and y are normally free-variable parameters in the class expression substituted for C, which can be thought of as C ( x , y ). See comments for abrexex 5724. (Contributed by NM, 20-Sep-2011.)
|- A e. _V   &    |- B e. _V   =>    |- { z | E. x e. A E. y e. B z = C } e. _V
Theorem	ab2rexex2 5962*	Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x, y, and z. Compare abrexex2 5741. (Contributed by NM, 20-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- { z | ph } e. _V   =>    |- { z | E. x e. A E. y e. B ph } e. _V
Theorem	oprssdm 5963*	Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.)
|- -. (/) e. S   &    |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S )   =>    |- ( S X. S ) C_ dom F
Theorem	ndmovg 5964	The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)
|- ( ( dom F = ( R X. S ) /\ -. ( A e. R /\ B e. S ) ) -> ( A F B ) = (/) )
Theorem	ndmov 5965	The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.)
|- dom F = ( S X. S )   =>    |- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = (/) )
Theorem	ndmovcl 5966	The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.)
|- dom F = ( S X. S )   &    |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S )   &    |- (/) e. S   =>    |- ( A F B ) e. S
Theorem	ndmovrcl 5967	Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)
|- dom F = ( S X. S )   &    |- -. (/) e. S   =>    |- ( ( A F B ) e. S -> ( A e. S /\ B e. S ) )
Theorem	ndmovcom 5968	Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
|- dom F = ( S X. S )   =>    |- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = ( B F A ) )
Theorem	ndmovass 5969	Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
|- dom F = ( S X. S )   &    |- -. (/) e. S   =>    |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
Theorem	ndmovdistr 5970	Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
|- dom F = ( S X. S )   &    |- -. (/) e. S   &    |- dom G = ( S X. S )   =>    |- ( -. ( A e. S /\ B e. S /\ C e. S ) -> ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) )
Theorem	ndmovord 5971	Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)
|- dom F = ( S X. S )   &    |- R C_ ( S X. S )   &    |- -. (/) e. S   &    |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A R B <-> ( C F A ) R ( C F B ) ) )   =>    |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )
Theorem	ndmovordi 5972	Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.)
|- dom F = ( S X. S )   &    |- R C_ ( S X. S )   &    |- -. (/) e. S   &    |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) )   =>    |- ( ( C F A ) R ( C F B ) -> A R B )
Theorem	caovclg 5973*	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 5974*	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 5975*	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 5976*	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 5977*	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 5978*	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 5979*	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 5980*	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 5981*	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 5982*	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 5983*	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 5984*	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 5985*	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 5986*	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 5987*	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 5988*	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 5989*	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 5990*	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 5991*	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 5992*	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 5993*	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 5994*	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 5995*	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 5996*	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 5997*	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 5998*	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 5999*	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 6000*	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 ) )