HomeHome Intuitionistic Logic Explorer
Theorem List (p. 59 of 135)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoveqan12rd 5801 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ps 
 /\  ph )  ->  ( A F C )  =  ( B F D ) )
 
Theoremoveq123d 5802 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A F C )  =  ( B G D ) )
 
Theoremfvoveq1d 5803 Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  ( A O C ) )  =  ( F `  ( B O C ) ) )
 
Theoremfvoveq1 5804 Equality theorem for nested function and operation value. Closed form of fvoveq1d 5803. (Contributed by AV, 23-Jul-2022.)
 |-  ( A  =  B  ->  ( F `  ( A O C ) )  =  ( F `  ( B O C ) ) )
 
Theoremovanraleqv 5805* Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
 |-  ( B  =  X  ->  ( ph  <->  ps ) )   =>    |-  ( B  =  X  ->  ( A. x  e.  V  ( ph  /\  ( A  .x.  B )  =  C )  <->  A. x  e.  V  ( ps  /\  ( A 
 .x.  X )  =  C ) ) )
 
Theoremimbrov2fvoveq 5806 Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
 |-  ( X  =  Y  ->  ( ph  <->  ps ) )   =>    |-  ( X  =  Y  ->  ( ( ph  ->  ( F `  (
 ( G `  X )  .x.  O ) ) R A )  <->  ( ps  ->  ( F `  ( ( G `  Y ) 
 .x.  O ) ) R A ) ) )
 
Theoremnfovd 5807 Deduction version of bound-variable hypothesis builder nfov 5808. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x F )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x ( A F B ) )
 
Theoremnfov 5808 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x ( A F B )
 
Theoremoprabidlem 5809* Slight elaboration of exdistrfor 1773. A lemma for oprabid 5810. (Contributed by Jim Kingdon, 15-Jan-2019.)
 |-  ( E. x E. y ( x  =  z  /\  ps )  ->  E. x ( x  =  z  /\  E. y ps ) )
 
Theoremoprabid 5810 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between  x,  y, and  z, we use ax-bndl 1487 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( <. <. x ,  y >. ,  z >.  e.  { <.
 <. x ,  y >. ,  z >.  |  ph }  <->  ph )
 
Theoremfnovex 5811 The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
 |-  ( ( F  Fn  ( C  X.  D ) 
 /\  A  e.  C  /\  B  e.  D ) 
 ->  ( A F B )  e.  _V )
 
Theoremovexg 5812 Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
 |-  ( ( A  e.  V  /\  F  e.  W  /\  B  e.  X ) 
 ->  ( A F B )  e.  _V )
 
Theoremovprc 5813 The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 Rel  dom  F   =>    |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
 
Theoremovprc1 5814 The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
 |- 
 Rel  dom  F   =>    |-  ( -.  A  e.  _V 
 ->  ( A F B )  =  (/) )
 
Theoremovprc2 5815 The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 Rel  dom  F   =>    |-  ( -.  B  e.  _V 
 ->  ( A F B )  =  (/) )
 
Theoremcsbov123g 5816 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( A  e.  D  -> 
 [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) )
 
Theoremcsbov12g 5817* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C ) )
 
Theoremcsbov1g 5818* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F C ) )
 
Theoremcsbov2g 5819* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B F C )  =  ( B F [_ A  /  x ]_ C ) )
 
Theoremrspceov 5820* A frequently used special case of rspc2ev 2807 for operation values. (Contributed by NM, 21-Mar-2007.)
 |-  ( ( C  e.  A  /\  D  e.  B  /\  S  =  ( C F D ) ) 
 ->  E. x  e.  A  E. y  e.  B  S  =  ( x F y ) )
 
Theoremfnotovb 5821 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5470. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A  /\  D  e.  B ) 
 ->  ( ( C F D )  =  R  <->  <. C ,  D ,  R >.  e.  F ) )
 
Theoremopabbrex 5822* A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  ( ( V  e.  _V 
 /\  E  e.  _V )  ->  ( f ( V W E ) p  ->  th )
 )   &    |-  ( ( V  e.  _V 
 /\  E  e.  _V )  ->  { <. f ,  p >.  |  th }  e.  _V )   =>    |-  ( ( V  e.  _V 
 /\  E  e.  _V )  ->  { <. f ,  p >.  |  (
 f ( V W E ) p  /\  ps ) }  e.  _V )
 
Theorem0neqopab 5823 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
 |- 
 -.  (/)  e.  { <. x ,  y >.  |  ph }
 
Theorembrabvv 5824* If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.)
 |-  ( X { <. x ,  y >.  |  ph } Y  ->  ( X  e.  _V  /\  Y  e.  _V ) )
 
Theoremdfoprab2 5825* Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  = 
 <. x ,  y >.  /\  ph ) }
 
Theoremreloprab 5826* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
 |- 
 Rel  { <. <. x ,  y >. ,  z >.  |  ph }
 
Theoremnfoprab1 5827 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 5828 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 5829 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
 |-  F/_ z { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremnfoprab 5830* 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 5831* 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 5832* 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 5833* 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 5834 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4204. (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 5835 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4205. (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 5836 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4208. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  y >. ,  z >.  |  ps }  <->  A. x A. y A. z ( ph  <->  ps ) )
 
Theoremmpoeq123 5837* 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 5838* 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 5839* 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 5840* 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 5841 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 5842* 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 5843 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 5844* 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 5845 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 5846 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 5847* 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 5848 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 5849* 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 5850* 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 5851* 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 5852* 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 5853* 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 5854* 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 5855* 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 5856* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5857 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 5857* 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 5858* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4030, 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 5859* 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 5860* 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 5861* 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 5862* 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 5863* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
 |- 
 Rel  dom  { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremoprabss 5864* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph } 
 C_  ( ( _V 
 X.  _V )  X.  _V )
 
Theoremeloprabga 5865* The law of concretion for operation class abstraction. Compare elopab 4187. (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 5866* The law of concretion for operation class abstraction. Compare elopab 4187. (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 5867* 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 5868* 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 5869* 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 5870* 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 5871 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 5872 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 5873* 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 5874* 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 5875* 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 5876* 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 5877* "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 5878* "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 5879* 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 5880* 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 5881* 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 5882* 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 5883 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 5884* 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 5885* 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 5886* 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 5887* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5885. (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 5888* 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 5889* 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 5890* 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 5891* 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 5892* 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 5893* 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 5894* 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 5895* The value of an operation class abstraction. Compare ovidi 5896. 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 5896* 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 5897* 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 5898* The value of an operation class abstraction. Compare ovig 5899. 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 5899* 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 5900* Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5511.) (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 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13439
  Copyright terms: Public domain < Previous  Next >