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Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaovcan 5901* 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 5902* 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 5903* 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 5904* 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 5905* 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 5906* 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 5907* 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 5908* 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 ) ) )
 
Theoremcaovord2 5909* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( z  e.  S  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   =>    |-  ( C  e.  S  ->  ( A R B  <->  ( A F C ) R ( B F C ) ) )
 
Theoremcaovord3 5910* Ordering law. (Contributed by NM, 29-Feb-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( z  e.  S  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  D  e.  _V   =>    |-  (
 ( ( B  e.  S  /\  C  e.  S )  /\  ( A F B )  =  ( C F D ) ) 
 ->  ( A R C  <->  D R B ) )
 
Theoremcaovdig 5911* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
 |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )   =>    |-  ( ( ph  /\  ( A  e.  K  /\  B  e.  S  /\  C  e.  S )
 )  ->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
 
Theoremcaovdid 5912* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   =>    |-  ( ph  ->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
 
Theoremcaovdir2d 5913* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x G y )  =  ( y G x ) )   =>    |-  ( ph  ->  (
 ( A F B ) G C )  =  ( ( A G C ) F ( B G C ) ) )
 
Theoremcaovdirg 5914* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K )
 )  ->  ( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) ) )   =>    |-  ( ( ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  K )
 )  ->  ( ( A F B ) G C )  =  ( ( A G C ) H ( B G C ) ) )
 
Theoremcaovdird 5915* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K )
 )  ->  ( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) ) )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  K )   =>    |-  ( ph  ->  (
 ( A F B ) G C )  =  ( ( A G C ) H ( B G C ) ) )
 
Theoremcaovdi 5916* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )   =>    |-  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) )
 
Theoremcaov32d 5917* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ph  ->  (
 ( A F B ) F C )  =  ( ( A F C ) F B ) )
 
Theoremcaov12d 5918* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ph  ->  ( A F ( B F C ) )  =  ( B F ( A F C ) ) )
 
Theoremcaov31d 5919* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ph  ->  (
 ( A F B ) F C )  =  ( ( C F B ) F A ) )
 
Theoremcaov13d 5920* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   =>    |-  ( ph  ->  ( A F ( B F C ) )  =  ( C F ( B F A ) ) )
 
Theoremcaov4d 5921* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( ph  ->  D  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   =>    |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) ) )
 
Theoremcaov411d 5922* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( ph  ->  D  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   =>    |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( C F B ) F ( A F D ) ) )
 
Theoremcaov42d 5923* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( ph  ->  D  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   =>    |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( D F B ) ) )
 
Theoremcaov32 5924* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( ( A F B ) F C )  =  ( ( A F C ) F B )
 
Theoremcaov12 5925* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( A F ( B F C ) )  =  ( B F ( A F C ) )
 
Theoremcaov31 5926* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
 
Theoremcaov13 5927* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x F y )  =  ( y F x )   &    |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )   =>    |-  ( A F ( B F C ) )  =  ( C F ( B F A ) )
 
Theoremcaovdilemd 5928* Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x G y )  =  ( y G x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) G z )  =  ( ( x G z ) F ( y G z ) ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x G y )  e.  S )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ph  ->  D  e.  S )   &    |-  ( ph  ->  H  e.  S )   =>    |-  ( ph  ->  (
 ( ( A G C ) F ( B G D ) ) G H )  =  ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) )
 
Theoremcaovlem2d 5929* Rearrangement of expression involving multiplication ( G) and addition ( F). (Contributed by Jim Kingdon, 3-Jan-2020.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x G y )  =  ( y G x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) G z )  =  ( ( x G z ) F ( y G z ) ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x G y )  e.  S )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  S )   &    |-  ( ph  ->  D  e.  S )   &    |-  ( ph  ->  H  e.  S )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   =>    |-  ( ph  ->  (
 ( ( ( A G C ) F ( B G D ) ) G H ) F ( ( ( A G D ) F ( B G C ) ) G R ) )  =  ( ( A G ( ( C G H ) F ( D G R ) ) ) F ( B G ( ( C G R ) F ( D G H ) ) ) ) )
 
Theoremcaovimo 5930* Uniqueness of inverse element in commutative, associative operation with identity. The identity element is  B. (Contributed by Jim Kingdon, 18-Sep-2019.)
 |-  B  e.  S   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  =  ( y F x ) )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x F y ) F z )  =  ( x F ( y F z ) ) )   &    |-  ( x  e.  S  ->  ( x F B )  =  x )   =>    |-  ( A  e.  S  ->  E* w ( w  e.  S  /\  ( A F w )  =  B ) )
 
Theoremgrprinvlem 5931* Lemma for grprinvd 5932. (Contributed by NM, 9-Aug-2013.)
 |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ph  ->  O  e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  B  ( y  .+  x )  =  O )   &    |-  (
 ( ph  /\  ps )  ->  X  e.  B )   &    |-  ( ( ph  /\  ps )  ->  ( X  .+  X )  =  X )   =>    |-  ( ( ph  /\  ps )  ->  X  =  O )
 
Theoremgrprinvd 5932* Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
 |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ph  ->  O  e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  B  ( y  .+  x )  =  O )   &    |-  (
 ( ph  /\  ps )  ->  X  e.  B )   &    |-  ( ( ph  /\  ps )  ->  N  e.  B )   &    |-  ( ( ph  /\  ps )  ->  ( N  .+  X )  =  O )   =>    |-  ( ( ph  /\  ps )  ->  ( X  .+  N )  =  O )
 
Theoremgrpridd 5933* Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
 |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ph  ->  O  e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  B  ( y  .+  x )  =  O )   =>    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( x  .+  O )  =  x )
 
2.6.11  Maps-to notation
 
Theoremelmpocl 5934* If a two-parameter class is inhabited, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( X  e.  ( S F T ) 
 ->  ( S  e.  A  /\  T  e.  B ) )
 
Theoremelmpocl1 5935* If a two-parameter class is inhabited, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( X  e.  ( S F T ) 
 ->  S  e.  A )
 
Theoremelmpocl2 5936* If a two-parameter class is inhabited, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( X  e.  ( S F T ) 
 ->  T  e.  B )
 
Theoremelovmpo 5937* Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  D  =  ( a  e.  A ,  b  e.  B  |->  C )   &    |-  C  e.  _V   &    |-  ( ( a  =  X  /\  b  =  Y )  ->  C  =  E )   =>    |-  ( F  e.  ( X D Y )  <->  ( X  e.  A  /\  Y  e.  B  /\  F  e.  E ) )
 
Theoremf1ocnvd 5938* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  W )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  D  e.  X )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
 ) )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  ( y  e.  B  |->  D ) ) )
 
Theoremf1od 5939* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  W )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  D  e.  X )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
 ) )   =>    |-  ( ph  ->  F : A -1-1-onto-> B )
 
Theoremf1ocnv2d 5940* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  B )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  D  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  ( y  e.  B  |->  D ) ) )
 
Theoremf1o2d 5941* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  B )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  D  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  ( ph  ->  F : A -1-1-onto-> B )
 
Theoremf1opw2 5942* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5943 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  ( ph  ->  F : A -1-1-onto-> B )   &    |-  ( ph  ->  ( `' F " a )  e.  _V )   &    |-  ( ph  ->  ( F "
 b )  e.  _V )   =>    |-  ( ph  ->  (
 b  e.  ~P A  |->  ( F " b ) ) : ~P A -1-1-onto-> ~P B )
 
Theoremf1opw 5943* A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( F : A -1-1-onto-> B  ->  ( b  e.  ~P A  |->  ( F "
 b ) ) : ~P A -1-1-onto-> ~P B )
 
Theoremsuppssfv 5944* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ph  ->  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
 ) )  C_  L )   &    |-  ( ph  ->  ( F `  Y )  =  Z )   &    |-  ( ( ph  /\  x  e.  D ) 
 ->  A  e.  V )   =>    |-  ( ph  ->  ( `' ( x  e.  D  |->  ( F `  A ) ) " ( _V  \  { Z } )
 )  C_  L )
 
Theoremsuppssov1 5945* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ph  ->  ( `' ( x  e.  D  |->  A ) " ( _V  \  { Y }
 ) )  C_  L )   &    |-  ( ( ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )   &    |-  ( ( ph  /\  x  e.  D ) 
 ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  D )  ->  B  e.  R )   =>    |-  ( ph  ->  ( `' ( x  e.  D  |->  ( A O B ) ) " ( _V  \  { Z } )
 )  C_  L )
 
2.6.12  Function operation
 
Syntaxcof 5946 Extend class notation to include mapping of an operation to a function operation.
 class  oF R
 
Syntaxcofr 5947 Extend class notation to include mapping of a binary relation to a function relation.
 class  oR R
 
Definitiondf-of 5948* Define the function operation map. The definition is designed so that if  R is a binary operation, then  oF R is the analogous operation on functions which corresponds to applying  R pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g
 )  |->  ( ( f `
  x ) R ( g `  x ) ) ) )
 
Definitiondf-ofr 5949* Define the function relation map. The definition is designed so that if  R is a binary relation, then  oF R is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  oR R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i 
 dom  g ) ( f `  x ) R ( g `  x ) }
 
Theoremofeq 5950 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( R  =  S  ->  oF R  =  oF S )
 
Theoremofreq 5951 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( R  =  S  ->  oR R  =  oR S )
 
Theoremofexg 5952 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
 |-  ( A  e.  V  ->  (  oF R  |`  A )  e.  _V )
 
Theoremnfof 5953 Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  oF R
 
Theoremnfofr 5954 Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  oR R
 
Theoremoffval 5955* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  C )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( G `  x )  =  D )   =>    |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( C R D ) ) )
 
Theoremofrfval 5956* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  C )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( G `  x )  =  D )   =>    |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  C R D ) )
 
Theoremofvalg 5957 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  X  e.  A )  ->  ( F `
  X )  =  C )   &    |-  ( ( ph  /\  X  e.  B ) 
 ->  ( G `  X )  =  D )   &    |-  (
 ( ph  /\  X  e.  S )  ->  ( C R D )  e.  U )   =>    |-  ( ( ph  /\  X  e.  S )  ->  (
 ( F  oF R G ) `  X )  =  ( C R D ) )
 
Theoremofrval 5958 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   &    |-  (
 ( ph  /\  X  e.  A )  ->  ( F `
  X )  =  C )   &    |-  ( ( ph  /\  X  e.  B ) 
 ->  ( G `  X )  =  D )   =>    |-  (
 ( ph  /\  F  oR R G  /\  X  e.  S )  ->  C R D )
 
Theoremofmresval 5959 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
 |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F (  oF R  |`  ( A  X.  B ) ) G )  =  ( F  oF R G ) )
 
Theoremoff 5960* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  T )
 )  ->  ( x R y )  e.  U )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : B
 --> T )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( F  oF R G ) : C --> U )
 
Theoremoffeq 5961* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  T )
 )  ->  ( x R y )  e.  U )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : B
 --> T )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  C   &    |-  ( ph  ->  H : C --> U )   &    |-  ( ( ph  /\  x  e.  A )  ->  ( F `  x )  =  D )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( G `  x )  =  E )   &    |-  (
 ( ph  /\  x  e.  C )  ->  ( D R E )  =  ( H `  x ) )   =>    |-  ( ph  ->  ( F  oF R G )  =  H )
 
Theoremofres 5962 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( F  oF R G )  =  ( ( F  |`  C )  oF R ( G  |`  C )
 ) )
 
Theoremoffval2 5963* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  X )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )   =>    |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( B R C ) ) )
 
Theoremofrfval2 5964* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  X )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )   =>    |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  A  B R C ) )
 
Theoremsuppssof1 5965* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ph  ->  ( `' A " ( _V  \  { Y } )
 )  C_  L )   &    |-  (
 ( ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )   &    |-  ( ph  ->  A : D --> V )   &    |-  ( ph  ->  B : D
 --> R )   &    |-  ( ph  ->  D  e.  W )   =>    |-  ( ph  ->  ( `' ( A  oF O B ) " ( _V  \  { Z }
 ) )  C_  L )
 
Theoremofco 5966 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  H : D --> C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( A  i^i  B )  =  C   =>    |-  ( ph  ->  ( ( F  oF R G )  o.  H )  =  ( ( F  o.  H )  oF R ( G  o.  H ) ) )
 
Theoremoffveqb 5967* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  ( ph  ->  H  Fn  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( F `  x )  =  B )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( G `  x )  =  C )   =>    |-  ( ph  ->  ( H  =  ( F  oF R G )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
 
Theoremofc12 5968 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  oF R ( A  X.  { C } ) )  =  ( A  X.  { ( B R C ) } ) )
 
Theoremcaofref 5969* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ( ph  /\  x  e.  S )  ->  x R x )   =>    |-  ( ph  ->  F  oR R F )
 
Theoremcaofinvl 5970* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  N : S --> S )   &    |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `  ( F `
  v ) ) ) )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( ( N `  x ) R x )  =  B )   =>    |-  ( ph  ->  ( G  oF R F )  =  ( A  X.  { B } ) )
 
Theoremcaofcom 5971* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x R y )  =  ( y R x ) )   =>    |-  ( ph  ->  ( F  oF R G )  =  ( G  oF R F ) )
 
Theoremcaofrss 5972* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x R y  ->  x T y ) )   =>    |-  ( ph  ->  ( F  oR R G  ->  F  oR T G ) )
 
Theoremcaoftrn 5973* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y  /\  y T z )  ->  x U z ) )   =>    |-  ( ph  ->  ( ( F  oR R G  /\  G  oR T H )  ->  F  oR U H ) )
 
2.6.13  Functions (continued)
 
TheoremresfunexgALT 5974 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5607 but requires ax-pow 4066 and ax-un 4323. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
Theoremcofunexg 5975 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  o.  B )  e.  _V )
 
Theoremcofunex2g 5976 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
 |-  ( ( A  e.  V  /\  Fun  `' B )  ->  ( A  o.  B )  e.  _V )
 
TheoremfnexALT 5977 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5175. This version of fnex 5608 uses ax-pow 4066 and ax-un 4323, whereas fnex 5608 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
 
Theoremfunrnex 5978 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5609. (Contributed by NM, 11-Nov-1995.)
 |-  ( dom  F  e.  B  ->  ( Fun  F  ->  ran  F  e.  _V ) )
 
Theoremfornex 5979 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  e.  _V ) )
 
Theoremf1dmex 5980 If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)
 |-  ( ( F : A -1-1-> B  /\  B  e.  C )  ->  A  e.  _V )
 
Theoremabrexex 5981* Existence of a class abstraction of existentially restricted sets.  x is normally a free-variable parameter in the class expression substituted for  B, which can be thought of as  B ( x ). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5611, funex 5609, fnex 5608, resfunexg 5607, and funimaexg 5175. See also abrexex2 5988. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   =>    |-  { y  | 
 E. x  e.  A  y  =  B }  e.  _V
 
Theoremabrexexg 5982* Existence of a class abstraction of existentially restricted sets.  x is normally a free-variable parameter in  B. The antecedent assures us that  A is a set. (Contributed by NM, 3-Nov-2003.)
 |-  ( A  e.  V  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
 
Theoremiunexg 5983* The existence of an indexed union. 
x is normally a free-variable parameter in  B. (Contributed by NM, 23-Mar-2006.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  W )  ->  U_ x  e.  A  B  e.  _V )
 
Theoremabrexex2g 5984* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  { y  | 
 ph }  e.  W )  ->  { y  | 
 E. x  e.  A  ph
 }  e.  _V )
 
Theoremopabex3d 5985* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  { y  |  ps }  e.  _V )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ( x  e.  A  /\  ps ) }  e.  _V )
 
Theoremopabex3 5986* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  { y  |  ph }  e.  _V )   =>    |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremiunex 5987* The existence of an indexed union. 
x is normally a free-variable parameter in the class expression substituted for  B, which can be read informally as  B ( x ). (Contributed by NM, 13-Oct-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  U_ x  e.  A  B  e.  _V
 
Theoremabrexex2 5988* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 5981. (Contributed by NM, 12-Sep-2004.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x  e.  A  ph
 }  e.  _V
 
Theoremabexssex 5989* Existence of a class abstraction with an existentially quantified expression. Both  x and  y can be free in  ph. (Contributed by NM, 29-Jul-2006.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x ( x 
 C_  A  /\  ph ) }  e.  _V
 
Theoremabexex 5990* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
 |-  A  e.  _V   &    |-  ( ph  ->  x  e.  A )   &    |- 
 { y  |  ph }  e.  _V   =>    |- 
 { y  |  E. x ph }  e.  _V
 
Theoremoprabexd 5991* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  E* z ps )   &    |-  ( ph  ->  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  A  /\  y  e.  B )  /\  ps ) }
 )   =>    |-  ( ph  ->  F  e.  _V )
 
Theoremoprabex 5992* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  E* z ph )   &    |-  F  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  A  /\  y  e.  B )  /\  ph ) }   =>    |-  F  e.  _V
 
Theoremoprabex3 5993* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
 |-  H  e.  _V   &    |-  F  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( H  X.  H )  /\  y  e.  ( H  X.  H ) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. ) 
 /\  z  =  R ) ) }   =>    |-  F  e.  _V
 
Theoremoprabrexex2 5994* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  A  e.  _V   &    |-  { <. <. x ,  y >. ,  z >.  |  ph }  e.  _V   =>    |-  {
 <. <. x ,  y >. ,  z >.  |  E. w  e.  A  ph }  e.  _V
 
Theoremab2rexex 5995* Existence of a class abstraction of existentially restricted sets. Variables  x and  y are normally free-variable parameters in the class expression substituted for  C, which can be thought of as  C ( x ,  y ). See comments for abrexex 5981. (Contributed by NM, 20-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  e.  _V
 
Theoremab2rexex2 5996* Existence of an existentially restricted class abstraction.  ph normally has free-variable parameters  x,  y, and  z. Compare abrexex2 5988. (Contributed by NM, 20-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  { z  | 
 ph }  e.  _V   =>    |-  { z  |  E. x  e.  A  E. y  e.  B  ph
 }  e.  _V
 
TheoremxpexgALT 5997 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4621 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  e.  _V )
 
Theoremoffval3 5998* General value of  ( F  oF R G ) with no assumptions on functionality of  F and  G. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
  x ) R ( G `  x ) ) ) )
 
Theoremoffres 5999 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  oF R G )  |`  D )  =  ( ( F  |`  D )  oF R ( G  |`  D )
 ) )
 
Theoremofmres 6000* Equivalent expressions for a restriction of the function operation map. Unlike  oF R which is a proper class,  (  oF R  |`  ( A  X.  B
) ) can be a set by ofmresex 6001, allowing it to be used as a function or structure argument. By ofmresval 5959, the restricted operation map values are the same as the original values, allowing theorems for  oF R to be reused. (Contributed by NM, 20-Oct-2014.)
 |-  (  oF R  |`  ( A  X.  B ) )  =  (
 f  e.  A ,  g  e.  B  |->  ( f  oF R g ) )
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