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Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaovdig 5801* 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 5802* 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 5803* 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 5804* 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 5805* 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 5806* 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 5807* 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 5808* 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 5809* 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 5810* 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 5811* 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 5812* 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 5813* 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 5814* 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 5815* 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 5816* 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 5817* 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 5818* 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 5819* 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 5820* 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 5821* Lemma for grprinvd 5822. (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 5822* 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 5823* 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
 
Theoremelmpt2cl 5824* If a two-parameter class is not empty, 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 ) )
 
Theoremelmpt2cl1 5825* If a two-parameter class is not empty, 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 )
 
Theoremelmpt2cl2 5826* If a two-parameter class is not empty, 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 )
 
Theoremelovmpt2 5827* 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 5828* 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 5829* 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 5830* 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 5831* 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 5832* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5833 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 5833* 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 5834* 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 5835* 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 5836 Extend class notation to include mapping of an operation to a function operation.
 class  oF R
 
Syntaxcofr 5837 Extend class notation to include mapping of a binary relation to a function relation.
 class  oR R
 
Definitiondf-of 5838* 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 5839* 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 5840 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( R  =  S  ->  oF R  =  oF S )
 
Theoremofreq 5841 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( R  =  S  ->  oR R  =  oR S )
 
Theoremofexg 5842 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 5843 Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  oF R
 
Theoremnfofr 5844 Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  oR R
 
Theoremoffval 5845* 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 5846* 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 ) )
 
Theoremfnofval 5847 Evaluate a function operation at a point. (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  ->  R  Fn  ( U  X.  V ) )   &    |-  ( ph  ->  C  e.  U )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ( ph  /\  X  e.  S ) 
 ->  ( ( F  oF R G ) `  X )  =  ( C R D ) )
 
Theoremofrval 5848 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 5849 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 5850* 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 )
 
Theoremofres 5851 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 5852* 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 5853* 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 5854* 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 5855 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 5856* 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 5857 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 5858* 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 5859* 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 5860* 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 5861* 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 5862* 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 5863 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 5500 but requires ax-pow 4001 and ax-un 4251. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
Theoremcofunexg 5864 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 5865 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 5866 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 5084. This version of fnex 5501 uses ax-pow 4001 and ax-un 4251, whereas fnex 5501 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 5867 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 5502. (Contributed by NM, 11-Nov-1995.)
 |-  ( dom  F  e.  B  ->  ( Fun  F  ->  ran  F  e.  _V ) )
 
Theoremfornex 5868 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 5869 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 5870* 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 5504, funex 5502, fnex 5501, resfunexg 5500, and funimaexg 5084. See also abrexex2 5877. (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 5871* 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 5872* 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 5873* 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 5874* 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 5875* 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 5876* 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 5877* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 5870. (Contributed by NM, 12-Sep-2004.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x  e.  A  ph
 }  e.  _V
 
Theoremabexssex 5878* 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 5879* 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 5880* 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 5881* 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 5882* 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 5883* 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 5884* 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 5870. (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 5885* Existence of an existentially restricted class abstraction.  ph normally has free-variable parameters  x,  y, and  z. Compare abrexex2 5877. (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 5886 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4540 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 5887* 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 5888 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 5889* 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 5890, allowing it to be used as a function or structure argument. By ofmresval 5849, 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 ) )
 
Theoremofmresex 5890 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  (  oF R  |`  ( A  X.  B ) )  e.  _V )
 
2.6.14  First and second members of an ordered pair
 
Syntaxc1st 5891 Extend the definition of a class to include the first member an ordered pair function.
 class  1st
 
Syntaxc2nd 5892 Extend the definition of a class to include the second member an ordered pair function.
 class  2nd
 
Definitiondf-1st 5893 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 5899 proves that it does this. For example, ( 1st `  <. 3 , 4  >.) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 4899 and op1stb 4290). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 1st  =  ( x  e.  _V  |->  U. dom  { x } )
 
Definitiondf-2nd 5894 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 5900 proves that it does this. For example,  ( 2nd ` 
<. 3 , 4 
>.) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 4902 and op2ndb 4901). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 2nd  =  ( x  e.  _V  |->  U. ran  { x } )
 
Theorem1stvalg 5895 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  _V  ->  ( 1st `  A )  =  U. dom  { A } )
 
Theorem2ndvalg 5896 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  _V  ->  ( 2nd `  A )  =  U. ran  { A } )
 
Theorem1st0 5897 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 1st `  (/) )  =  (/)
 
Theorem2nd0 5898 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 2nd `  (/) )  =  (/)
 
Theoremop1st 5899 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 1st `  <. A ,  B >. )  =  A
 
Theoremop2nd 5900 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 2nd `  <. A ,  B >. )  =  B
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