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Theorem List for Intuitionistic Logic Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfmpt2 5601* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
 |-  F/_ z A   &    |-  F/_ z B   &    |-  F/_ z C   =>    |-  F/_ z ( x  e.  A ,  y  e.  B  |->  C )
 
Theoremmpt20 5602 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)
 
Theoremoprab4 5603* Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }
 
Theoremcbvoprab1 5604* Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
 |- 
 F/ w ph   &    |-  F/ x ps   &    |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. w ,  y >. ,  z >.  |  ps }
 
Theoremcbvoprab2 5605* Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |- 
 F/ w ph   &    |-  F/ y ps   &    |-  ( y  =  w  ->  ( ph  <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  w >. ,  z >.  |  ps }
 
Theoremcbvoprab12 5606* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |- 
 F/ w ph   &    |-  F/ v ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph 
 <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. w ,  v >. ,  z >.  |  ps }
 
Theoremcbvoprab12v 5607* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)
 |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph 
 <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. w ,  v >. ,  z >.  |  ps }
 
Theoremcbvoprab3 5608* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
 |- 
 F/ w ph   &    |-  F/ z ps   &    |-  ( z  =  w  ->  ( ph  <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  y >. ,  w >.  |  ps }
 
Theoremcbvoprab3v 5609* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( z  =  w 
 ->  ( ph  <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  y >. ,  w >.  |  ps }
 
Theoremcbvmpt2x 5610* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5611 allows  B to be a function of  x. (Contributed by NM, 29-Dec-2014.)
 |-  F/_ z B   &    |-  F/_ x D   &    |-  F/_ z C   &    |-  F/_ w C   &    |-  F/_ x E   &    |-  F/_ y E   &    |-  ( x  =  z 
 ->  B  =  D )   &    |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  E )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  A ,  w  e.  D  |->  E )
 
Theoremcbvmpt2 5611* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
 |-  F/_ z C   &    |-  F/_ w C   &    |-  F/_ x D   &    |-  F/_ y D   &    |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  A ,  w  e.  B  |->  D )
 
Theoremcbvmpt2v 5612* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3879, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
 |-  ( x  =  z 
 ->  C  =  E )   &    |-  ( y  =  w  ->  E  =  D )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  A ,  w  e.  B  |->  D )
 
Theoremdmoprab 5613* The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |- 
 dom  { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. x ,  y >.  |  E. z ph }
 
Theoremdmoprabss 5614* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theoremrnoprab 5615* The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
 |- 
 ran  { <. <. x ,  y >. ,  z >.  |  ph }  =  { z  | 
 E. x E. y ph }
 
Theoremrnoprab2 5616* The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
 |- 
 ran  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  =  { z  |  E. x  e.  A  E. y  e.  B  ph }
 
Theoremreldmoprab 5617* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
 |- 
 Rel  dom  { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremoprabss 5618* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph } 
 C_  ( ( _V 
 X.  _V )  X.  _V )
 
Theoremeloprabga 5619* The law of concretion for operation class abstraction. Compare elopab 4023. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y >. ,  z >.  |  ph }  <->  ps ) )
 
Theoremeloprabg 5620* The law of concretion for operation class abstraction. Compare elopab 4023. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  th ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y >. ,  z >.  |  ph }  <->  th ) )
 
Theoremssoprab2i 5621* Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( ph  ->  ps )   =>    |-  { <. <. x ,  y >. ,  z >.  |  ph }  C_  {
 <. <. x ,  y >. ,  z >.  |  ps }
 
Theoremmpt2v 5622* Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
 |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <.
 <. x ,  y >. ,  z >.  |  z  =  C }
 
Theoremmpt2mptx 5623* Express a two-argument function as a one-argument function, or vice-versa. In this version 
B ( x ) is not assumed to be constant w.r.t  x. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( z  =  <. x ,  y >.  ->  C  =  D )   =>    |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
 
Theoremmpt2mpt 5624* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
 |-  ( z  =  <. x ,  y >.  ->  C  =  D )   =>    |-  ( z  e.  ( A  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
 
Theoremresoprab 5625* 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 ) }
 
Theoremresoprab2 5626* 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 ) } )
 
Theoremresmpt2 5627* 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 ) )
 
Theoremfunoprabg 5628* "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 } )
 
Theoremfunoprab 5629* "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 }
 
Theoremfnoprabg 5630* 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 } )
 
Theoremmpt2fun 5631* 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
 
Theoremfnoprab 5632* 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 }
 
Theoremffnov 5633* 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 ) )
 
Theoremfovcl 5634 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 )
 
Theoremeqfnov 5635* 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 ) ) ) )
 
Theoremeqfnov2 5636* 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 ) ) )
 
Theoremfnovim 5637* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 16-Jan-2019.)
 |-  ( F  Fn  ( A  X.  B )  ->  F  =  ( x  e.  A ,  y  e.  B  |->  ( x F y ) ) )
 
Theoremmpt22eqb 5638* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5636. (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 ) )
 
Theoremrnmpt2 5639* 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 }
 
Theoremreldmmpt2 5640* 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
 
Theoremelrnmpt2g 5641* 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 )
 )
 
Theoremelrnmpt2 5642* 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 )
 
Theoremralrnmpt2 5643* 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 ) )
 
Theoremrexrnmpt2 5644* 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 ) )
 
Theoremovid 5645* 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 ) )
 
Theoremovidig 5646* The value of an operation class abstraction. Compare ovidi 5647. 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 )
 
Theoremovidi 5647* 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 ) )
 
Theoremov 5648* 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 ) )
 
Theoremovigg 5649* The value of an operation class abstraction. Compare ovig 5650. 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 )
 )
 
Theoremovig 5650* 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 )
 )
 
Theoremovmpt4g 5651* Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5282.) (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 )
 
Theoremovmpt2s 5652* 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 )
 
Theoremov2gf 5653* The value of an operation class abstraction. A version of ovmpt2g 5663 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 )
 
Theoremovmpt2dxf 5654* 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 )
 
Theoremovmpt2dx 5655* 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 )
 
Theoremovmpt2d 5656* 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 )
 
Theoremovmpt2x 5657* The value of an operation class abstraction. Variant of ovmpt2ga 5658 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 )
 
Theoremovmpt2ga 5658* 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 )
 
Theoremovmpt2a 5659* 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 )
 
Theoremovmpt2df 5660* Alternate deduction version of ovmpt2 5664, 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 ) )
 
Theoremovmpt2dv 5661* Alternate deduction version of ovmpt2 5664, 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 ) )
 
Theoremovmpt2dv2 5662* Alternate deduction version of ovmpt2 5664, 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 ) )
 
Theoremovmpt2g 5663* 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 )
 
Theoremovmpt2 5664* 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 )
 
Theoremovi3 5665* 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 )
 
Theoremov6g 5666* 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 )
 
Theoremovg 5667* 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 ) )
 
Theoremovres 5668 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 ) )
 
Theoremovresd 5669 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 ) )
 
Theoremoprssov 5670 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 ) )
 
Theoremfovrn 5671 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 )
 
Theoremfovrnda 5672 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 )
 
Theoremfovrnd 5673 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 )
 
Theoremfnrnov 5674* 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 ) }
 )
 
Theoremfoov 5675* 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 ) ) )
 
Theoremfnovrn 5676 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 )
 
Theoremovelrn 5677* 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 ) ) )
 
Theoremfunimassov 5678* 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 ) )
 
Theoremovelimab 5679* 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 ) ) )
 
Theoremovconst2 5680 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 )
 
Theoremcaovclg 5681* 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 )
 
Theoremcaovcld 5682* 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 )
 
Theoremcaovcl 5683* 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 )
 
Theoremcaovcomg 5684* 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 ) )
 
Theoremcaovcomd 5685* 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 ) )
 
Theoremcaovcom 5686* 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 )
 
Theoremcaovassg 5687* 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 ) ) )
 
Theoremcaovassd 5688* 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 ) ) )
 
Theoremcaovass 5689* 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 ) )
 
Theoremcaovcang 5690* 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 ) )
 
Theoremcaovcand 5691* 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 ) )
 
Theoremcaovcanrd 5692* 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 ) )
 
Theoremcaovcan 5693* 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 ) )
 
Theoremcaovordig 5694* 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 ) ) )
 
Theoremcaovordid 5695* 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 ) ) )
 
Theoremcaovordg 5696* 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 ) ) )
 
Theoremcaovordd 5697* 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 ) ) )
 
Theoremcaovord2d 5698* 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 ) ) )
 
Theoremcaovord3d 5699* 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 ) ) )
 
Theoremcaovord 5700* 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 ) ) )
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