P. 61 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	caovassd 6001*	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 6002*	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 6003*	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 6004*	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 6005*	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 6006*	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 6007*	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 6008*	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 6009*	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 6010*	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 6011*	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 6012*	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 6013*	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 6014*	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 6015*	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 6016*	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 6017*	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 6018*	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 6019*	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 6020*	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 6021*	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 6022*	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 6023*	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 6024*	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 6025*	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 6026*	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 6027*	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 6028*	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 6029*	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 6030*	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 6031*	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 6032*	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 6033*	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 6034*	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 6035*	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 ) )
2.6.12  Maps-to notation
Theorem	elmpocl 6036*	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 6037*	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 6038*	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 6039*	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 6040*	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 6041*	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 6042*	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 6043*	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 6044*	A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6045 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 6045*	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 6046*	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 6047*	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 )
2.6.13  Function operation
Syntax	cof 6048	Extend class notation to include mapping of an operation to a function operation.
class oF R
Syntax	cofr 6049	Extend class notation to include mapping of a binary relation to a function relation.
class oR R
Definition	df-of 6050*	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 6051*	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 6052	Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( R = S -> oF R = oF S )
Theorem	ofreq 6053	Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( R = S -> oR R = oR S )
Theorem	ofexg 6054	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 6055	Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- F/_ x R   =>    |- F/_ x oF R
Theorem	nfofr 6056	Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- F/_ x R   =>    |- F/_ x oR R
Theorem	offval 6057*	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 6058*	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 6059	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 6060	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 6061	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 6062*	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 6063*	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 6064	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 6065*	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 6066*	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 6067*	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 6068	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 ) ) )
Theorem	offveqb 6069*	Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
|- ( ph -> A e. V )   &    |- ( ph -> F Fn A )   &    |- ( ph -> G Fn A )   &    |- ( ph -> H Fn A )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )   &    |- ( ( ph /\ x e. A ) -> ( G ` x ) = C )   =>    |- ( ph -> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )
Theorem	ofc12 6070	Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   =>    |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )
Theorem	caofref 6071*	Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ( ph /\ x e. S ) -> x R x )   =>    |- ( ph -> F oR R F )
Theorem	caofinvl 6072*	Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> B e. W )   &    |- ( ph -> N : S --> S )   &    |- ( ph -> G = ( v e. A |-> ( N ` ( F ` v ) ) ) )   &    |- ( ( ph /\ x e. S ) -> ( ( N ` x ) R x ) = B )   =>    |- ( ph -> ( G oF R F ) = ( A X. { B } ) )
Theorem	caofcom 6073*	Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : A --> S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y ) = ( y R x ) )   =>    |- ( ph -> ( F oF R G ) = ( G oF R F ) )
Theorem	caofrss 6074*	Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : A --> S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y -> x T y ) )   =>    |- ( ph -> ( F oR R G -> F oR T G ) )
Theorem	caoftrn 6075*	Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : A --> S )   &    |- ( ph -> H : A --> S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x R y /\ y T z ) -> x U z ) )   =>    |- ( ph -> ( ( F oR R G /\ G oR T H ) -> F oR U H ) )
2.6.14  Functions (continued)
Theorem	resfunexgALT 6076	The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5706 but requires ax-pow 4153 and ax-un 4411. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )
Theorem	cofunexg 6077	Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
|- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V )
Theorem	cofunex2g 6078	Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
|- ( ( A e. V /\ Fun `' B ) -> ( A o. B ) e. _V )
Theorem	fnexALT 6079	If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5272. This version of fnex 5707 uses ax-pow 4153 and ax-un 4411, whereas fnex 5707 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( F Fn A /\ A e. B ) -> F e. _V )
Theorem	funexw 6080	Weak version of funex 5708 that holds without ax-coll 4097. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
|- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )
Theorem	mptexw 6081*	Weak version of mptex 5711 that holds without ax-coll 4097. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
|- A e. _V   &    |- C e. _V   &    |- A. x e. A B e. C   =>    |- ( x e. A |-> B ) e. _V
Theorem	funrnex 6082	If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5708. (Contributed by NM, 11-Nov-1995.)
|- ( dom F e. B -> ( Fun F -> ran F e. _V ) )
Theorem	fornex 6083	If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
|- ( A e. C -> ( F : A -onto-> B -> B e. _V ) )
Theorem	f1dmex 6084	If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)
|- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V )
Theorem	abrexex 6085*	Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in the class expression substituted for B, which can be thought of as B ( x ). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5710, funex 5708, fnex 5707, resfunexg 5706, and funimaexg 5272. See also abrexex2 6092. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- A e. _V   =>    |- { y | E. x e. A y = B } e. _V
Theorem	abrexexg 6086*	Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in B. The antecedent assures us that A is a set. (Contributed by NM, 3-Nov-2003.)
|- ( A e. V -> { y | E. x e. A y = B } e. _V )
Theorem	iunexg 6087*	The existence of an indexed union. x is normally a free-variable parameter in B. (Contributed by NM, 23-Mar-2006.)
|- ( ( A e. V /\ A. x e. A B e. W ) -> U_ x e. A B e. _V )
Theorem	abrexex2g 6088*	Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( A e. V /\ A. x e. A { y | ph } e. W ) -> { y | E. x e. A ph } e. _V )
Theorem	opabex3d 6089*	Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
|- ( ph -> A e. _V )   &    |- ( ( ph /\ x e. A ) -> { y | ps } e. _V )   =>    |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )
Theorem	opabex3 6090*	Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. _V   &    |- ( x e. A -> { y | ph } e. _V )   =>    |- { <. x , y >. | ( x e. A /\ ph ) } e. _V
Theorem	iunex 6091*	The existence of an indexed union. x is normally a free-variable parameter in the class expression substituted for B, which can be read informally as B ( x ). (Contributed by NM, 13-Oct-2003.)
|- A e. _V   &    |- B e. _V   =>    |- U_ x e. A B e. _V
Theorem	abrexex2 6092*	Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. See also abrexex 6085. (Contributed by NM, 12-Sep-2004.)
|- A e. _V   &    |- { y | ph } e. _V   =>    |- { y | E. x e. A ph } e. _V
Theorem	abexssex 6093*	Existence of a class abstraction with an existentially quantified expression. Both x and y can be free in ph. (Contributed by NM, 29-Jul-2006.)
|- A e. _V   &    |- { y | ph } e. _V   =>    |- { y | E. x ( x C_ A /\ ph ) } e. _V
Theorem	abexex 6094*	A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
|- A e. _V   &    |- ( ph -> x e. A )   &    |- { y | ph } e. _V   =>    |- { y | E. x ph } e. _V
Theorem	oprabexd 6095*	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 6096*	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 6097*	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 6098*	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	ab2rexex 6099*	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 6085. (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 6100*	Existence of an existentially restricted class abstraction. ph normally has free-variable parameters x, y, and z. Compare abrexex2 6092. (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