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Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreloprab 5901* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
 |- 
 Rel  { <. <. x ,  y >. ,  z >.  |  ph }
 
Theoremnfoprab1 5902 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  F/_ x { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremnfoprab2 5903 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)
 |-  F/_ y { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremnfoprab3 5904 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
 |-  F/_ z { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremnfoprab 5905* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
 |- 
 F/ w ph   =>    |-  F/_ w { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremoprabbid 5906* Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |- 
 F/ x ph   &    |-  F/ y ph   &    |-  F/ z ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  {
 <. <. x ,  y >. ,  z >.  |  ps }  =  { <. <. x ,  y >. ,  z >.  |  ch } )
 
Theoremoprabbidv 5907* Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  {
 <. <. x ,  y >. ,  z >.  |  ps }  =  { <. <. x ,  y >. ,  z >.  |  ch } )
 
Theoremoprabbii 5908* Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( ph  <->  ps )   =>    |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z >.  |  ps }
 
Theoremssoprab2 5909 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4260. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( A. x A. y A. z ( ph  ->  ps )  ->  { <. <. x ,  y >. ,  z >.  |  ph }  C_  {
 <. <. x ,  y >. ,  z >.  |  ps } )
 
Theoremssoprab2b 5910 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4261. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( { <. <. x ,  y >. ,  z >.  | 
 ph }  C_  { <. <. x ,  y >. ,  z >.  |  ps }  <->  A. x A. y A. z ( ph  ->  ps ) )
 
Theoremeqoprab2b 5911 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4264. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  y >. ,  z >.  |  ps }  <->  A. x A. y A. z ( ph  <->  ps ) )
 
Theoremmpoeq123 5912* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F ) )
 
Theoremmpoeq12 5913* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  ( x  e.  A ,  y  e.  B  |->  E )  =  ( x  e.  C ,  y  e.  D  |->  E ) )
 
Theoremmpoeq123dva 5914* An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  =  E )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  C  =  F )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F ) )
 
Theoremmpoeq123dv 5915* An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ph  ->  B  =  E )   &    |-  ( ph  ->  C  =  F )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F ) )
 
Theoremmpoeq123i 5916 An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.)
 |-  A  =  D   &    |-  B  =  E   &    |-  C  =  F   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )
 
Theoremmpoeq3dva 5917* Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.)
 |-  ( ( ph  /\  x  e.  A  /\  y  e.  B )  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D ) )
 
Theoremmpoeq3ia 5918 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
 
Theoremmpoeq3dv 5919* An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.)
 |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D ) )
 
Theoremnfmpo1 5920 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
 |-  F/_ x ( x  e.  A ,  y  e.  B  |->  C )
 
Theoremnfmpo2 5921 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
 |-  F/_ y ( x  e.  A ,  y  e.  B  |->  C )
 
Theoremnfmpo 5922* 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 )
 
Theoremmpo0 5923 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 5924* 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 5925* 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 5926* 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 5927* 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 5928* 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 5929* 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 5930* 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 }
 
Theoremcbvmpox 5931* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5932 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 )
 
Theoremcbvmpo 5932* 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 )
 
Theoremcbvmpov 5933* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4084, 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 5934* 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 5935* 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 5936* 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 5937* 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 5938* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
 |- 
 Rel  dom  { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremoprabss 5939* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph } 
 C_  ( ( _V 
 X.  _V )  X.  _V )
 
Theoremeloprabga 5940* The law of concretion for operation class abstraction. Compare elopab 4243. (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 5941* The law of concretion for operation class abstraction. Compare elopab 4243. (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 5942* 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 }
 
Theoremmpov 5943* 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 }
 
Theoremmpomptx 5944* 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 )
 
Theoremmpompt 5945* 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 )
 
Theoremmpodifsnif 5946 A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.)
 |-  ( i  e.  ( A  \  { X }
 ) ,  j  e.  B  |->  if ( i  =  X ,  C ,  D ) )  =  ( i  e.  ( A  \  { X }
 ) ,  j  e.  B  |->  D )
 
Theoremmposnif 5947 A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.)
 |-  ( i  e.  { X } ,  j  e.  B  |->  if ( i  =  X ,  C ,  D ) )  =  ( i  e.  { X } ,  j  e.  B  |->  C )
 
Theoremfconstmpo 5948* Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)
 |-  ( ( A  X.  B )  X.  { C } )  =  ( x  e.  A ,  y  e.  B  |->  C )
 
Theoremresoprab 5949* 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 5950* 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 ) } )
 
Theoremresmpo 5951* 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 5952* "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 5953* "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 5954* 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 } )
 
Theoremmpofun 5955* 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 5956* 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 5957* 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 5958 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 5959* 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 5960* 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 5961* 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 ) ) )
 
Theoremmpo2eqb 5962* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5960. (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 ) )
 
Theoremrnmpo 5963* 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 }
 
Theoremreldmmpo 5964* 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
 
Theoremelrnmpog 5965* 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 )
 )
 
Theoremelrnmpo 5966* 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 )
 
Theoremralrnmpo 5967* 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 ) )
 
Theoremrexrnmpo 5968* 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 5969* 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 5970* The value of an operation class abstraction. Compare ovidi 5971. 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 5971* 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 5972* 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 5973* The value of an operation class abstraction. Compare ovig 5974. 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 5974* 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 5975* Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5579.) (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 )
 
Theoremovmpos 5976* 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 5977* The value of an operation class abstraction. A version of ovmpog 5987 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 )
 
Theoremovmpodxf 5978* 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 )
 
Theoremovmpodx 5979* 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 )
 
Theoremovmpod 5980* 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 )
 
Theoremovmpox 5981* The value of an operation class abstraction. Variant of ovmpoga 5982 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 )
 
Theoremovmpoga 5982* 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 )
 
Theoremovmpoa 5983* 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 )
 
Theoremovmpodf 5984* Alternate deduction version of ovmpo 5988, 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 ) )
 
Theoremovmpodv 5985* Alternate deduction version of ovmpo 5988, 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 ) )
 
Theoremovmpodv2 5986* Alternate deduction version of ovmpo 5988, 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 ) )
 
Theoremovmpog 5987* 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 )
 
Theoremovmpo 5988* 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 5989* 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 5990* 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 5991* 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 5992 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 5993 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 5994 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 5995 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 5996 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 5997 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 5998* 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 5999* 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 6000 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 )
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