HomeHome Intuitionistic Logic Explorer
Theorem List (p. 58 of 106)
< 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 - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaovord2 5701* 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 5702* 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 5703* 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 5704* 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 5705* 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 5706* 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 5707* 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 5708* 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 5709* 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 5710* 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 5711* 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 5712* 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 5713* 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 5714* 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 5715* 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 5716* 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 5717* 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 5718* 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 5719* 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 5720* 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 5721* 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 5722* 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 5723* Lemma for grprinvd 5724. (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 5724* 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 5725* 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 5726* 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 5727* 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 5728* 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 5729* 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 5730* 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 5731* 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 5732* 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 5733* 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 5734* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5735 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 5735* 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 5736* 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 5737* 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 5738 Extend class notation to include mapping of an operation to a function operation.
 class  oF R
 
Syntaxcofr 5739 Extend class notation to include mapping of a binary relation to a function relation.
 class  oR R
 
Definitiondf-of 5740* 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 5741* 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 5742 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( R  =  S  ->  oF R  =  oF S )
 
Theoremofreq 5743 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( R  =  S  ->  oR R  =  oR S )
 
Theoremofexg 5744 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 5745* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  oF R
 
Theoremnfofr 5746* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  oR R
 
Theoremoffval 5747* 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 5748* 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 5749 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 5750 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 5751 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 5752* 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 5753 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 5754* 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 5755* 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 5756* 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 5757 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 5758* 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 5759 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 5760* 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 5761* 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 5762* 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 5763* 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 5764* 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 5765 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 5410 but requires ax-pow 3955 and ax-un 4198. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
Theoremcofunexg 5766 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 5767 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 5768 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 5011. This version of fnex 5411 uses ax-pow 3955 and ax-un 4198, whereas fnex 5411 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 5769 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 5412. (Contributed by NM, 11-Nov-1995.)
 |-  ( dom  F  e.  B  ->  ( Fun  F  ->  ran  F  e.  _V ) )
 
Theoremfornex 5770 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 5771 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 5772* 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 5414, funex 5412, fnex 5411, resfunexg 5410, and funimaexg 5011. See also abrexex2 5779. (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 5773* 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 5774* 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 5775* 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 5776* 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 5777* 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 5778* 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 5779* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 5772. (Contributed by NM, 12-Sep-2004.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x  e.  A  ph
 }  e.  _V
 
Theoremabexssex 5780* 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 5781* 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 5782* 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 5783* 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 5784* 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 5785* 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 5786* 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 5772. (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 5787* Existence of an existentially restricted class abstraction.  ph normally has free-variable parameters  x,  y, and  z. Compare abrexex2 5779. (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 5788 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4480 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 5789* 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 5790 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 5791* 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 5792, allowing it to be used as a function or structure argument. By ofmresval 5751, 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 5792 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 5793 Extend the definition of a class to include the first member an ordered pair function.
 class  1st
 
Syntaxc2nd 5794 Extend the definition of a class to include the second member an ordered pair function.
 class  2nd
 
Definitiondf-1st 5795 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 5801 proves that it does this. For example, ( 1st `  <. 3 , 4  >.) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 4830 and op1stb 4237). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 1st  =  ( x  e.  _V  |->  U. dom  { x } )
 
Definitiondf-2nd 5796 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 5802 proves that it does this. For example,  ( 2nd ` 
<. 3 , 4 
>.) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 4833 and op2ndb 4832). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 2nd  =  ( x  e.  _V  |->  U. ran  { x } )
 
Theorem1stvalg 5797 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 5798 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 5799 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 1st `  (/) )  =  (/)
 
Theorem2nd0 5800 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 2nd `  (/) )  =  (/)
    < 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 10501-10511
  Copyright terms: Public domain < Previous  Next >