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Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1od 6001* 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 6002* 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 6003* 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 )
 
TheoremxpexgALT 6004 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4788 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  e.  _V )
 
Theoremf1opw2 6005* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6006 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 6006* 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 )
 
Theoremsuppss2 6007* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.)
 |-  ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )   =>    |-  ( ph  ->  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
 ) )  C_  W )
 
Theoremsuppssfv 6008* 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 6009* 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 )
 
2.4.10  Function operation
 
Syntaxcof 6010 Extend class notation to include mapping of an operation to a function operation.
 class  o F R
 
Syntaxcofr 6011 Extend class notation to include mapping of an binary relation to a function relation.
 class  o R R
 
Definitiondf-of 6012* Define the function operation map. The definition is designed so that if  R is a binary operation, then  o F 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.)
 |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g
 )  |->  ( ( f `
  x ) R ( g `  x ) ) ) )
 
Definitiondf-ofr 6013* Define the function relation map. The definition is designed so that if  R is a binary relation, then  o F 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.)
 |-  o R R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i 
 dom  g ) ( f `  x ) R ( g `  x ) }
 
Theoremofeq 6014 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  ( R  =  S  ->  o F R  =  o F S )
 
Theoremofreq 6015 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( R  =  S  ->  o R R  =  o R S )
 
Theoremofexg 6016 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
 |-  ( A  e.  V  ->  (  o F R  |`  A )  e.  _V )
 
Theoremnfof 6017* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  o F R
 
Theoremnfofr 6018* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  F/_ x R   =>    |-  F/_ x  o R R
 
Theoremoffval 6019* 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  o F R G )  =  ( x  e.  S  |->  ( C R D ) ) )
 
Theoremofrfval 6020* 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  o R R G  <->  A. x  e.  S  C R D ) )
 
Theoremofval 6021 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  /\  X  e.  S )  ->  ( ( F  o F R G ) `  X )  =  ( C R D ) )
 
Theoremofrval 6022 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  o R R G  /\  X  e.  S )  ->  C R D )
 
Theoremoffn 6023 The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( A  i^i  B )  =  S   =>    |-  ( ph  ->  ( F  o F R G )  Fn  S )
 
Theoremfnfvof 6024 Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)
 |-  ( ( ( F  Fn  A  /\  G  Fn  A )  /\  ( A  e.  V  /\  X  e.  A )
 )  ->  ( ( F  o F R G ) `  X )  =  ( ( F `  X ) R ( G `  X ) ) )
 
Theoremoffval3 6025* General value of  ( F  o F 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  o F R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
  x ) R ( G `  x ) ) ) )
 
Theoremoffres 6026 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  o F R G )  |`  D )  =  ( ( F  |`  D )  o F R ( G  |`  D )
 ) )
 
Theoremoff 6027* 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  o F R G ) : C --> U )
 
Theoremofres 6028 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  o F R G )  =  (
 ( F  |`  C )  o F R ( G  |`  C )
 ) )
 
Theoremoffval2 6029* 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  o F R G )  =  ( x  e.  A  |->  ( B R C ) ) )
 
Theoremofrfval2 6030* 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  o R R G 
 <-> 
 A. x  e.  A  B R C ) )
 
Theoremofco 6031 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  o F R G )  o.  H )  =  ( ( F  o.  H )  o F R ( G  o.  H ) ) )
 
Theoremoffveq 6032* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( 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  /\  x  e.  A ) 
 ->  ( B R C )  =  ( H `  x ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  H )
 
Theoremoffveqb 6033* 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  o F R G )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
 
Theoremofc1 6034 Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ( ph  /\  X  e.  A )  ->  ( F `  X )  =  C )   =>    |-  ( ( ph  /\  X  e.  A )  ->  (
 ( ( A  X.  { B } )  o F R F ) `
  X )  =  ( B R C ) )
 
Theoremofc2 6035 Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ( ph  /\  X  e.  A )  ->  ( F `  X )  =  C )   =>    |-  ( ( ph  /\  X  e.  A )  ->  (
 ( F  o F R ( A  X.  { B } ) ) `
  X )  =  ( C R B ) )
 
Theoremofc12 6036 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 } )  o F R ( A  X.  { C } ) )  =  ( A  X.  { ( B R C ) } ) )
 
Theoremcaofref 6037* 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  o R R F )
 
Theoremcaofinvl 6038* 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  o F R F )  =  ( A  X.  { B } ) )
 
Theoremcaofid0l 6039* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( B R x )  =  x )   =>    |-  ( ph  ->  ( ( A  X.  { B }
 )  o F R F )  =  F )
 
Theoremcaofid0r 6040* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( x R B )  =  x )   =>    |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  F )
 
Theoremcaofid1 6041* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  (
 ( ph  /\  x  e.  S )  ->  ( x R B )  =  C )   =>    |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  ( A  X.  { C } ) )
 
Theoremcaofid2 6042* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  (
 ( ph  /\  x  e.  S )  ->  ( B R x )  =  C )   =>    |-  ( ph  ->  (
 ( A  X.  { B } )  o F R F )  =  ( A  X.  { C } ) )
 
Theoremcaofcom 6043* 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  o F R G )  =  ( G  o F R F ) )
 
Theoremcaofrss 6044* 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  o R R G  ->  F  o R T G ) )
 
Theoremcaofass 6045* Transfer an associative 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  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y ) T z )  =  ( x O ( y P z ) ) )   =>    |-  ( ph  ->  (
 ( F  o F R G )  o F T H )  =  ( F  o F O ( G  o F P H ) ) )
 
Theoremcaoftrn 6046* 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  o R R G  /\  G  o R T H )  ->  F  o R U H ) )
 
Theoremcaofdi 6047* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> K )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )   =>    |-  ( ph  ->  ( F  o F T ( G  o F R H ) )  =  ( ( F  o F T G )  o F O ( F  o F T H ) ) )
 
Theoremcaofdir 6048* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> K )   &    |-  ( ph  ->  G : A
 --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K )
 )  ->  ( ( x R y ) T z )  =  ( ( x T z ) O ( y T z ) ) )   =>    |-  ( ph  ->  (
 ( G  o F R H )  o F T F )  =  ( ( G  o F T F )  o F O ( H  o F T F ) ) )
 
Theoremcaonncan 6049* Transfer nncan 9044-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  A : I --> S )   &    |-  ( ph  ->  B : I
 --> S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x M ( x M y ) )  =  y )   =>    |-  ( ph  ->  ( A  o F M ( A  o F M B ) )  =  B )
 
Theoremofmres 6050* Equivalent expressions for a restriction of the function operation map. Unlike  o F R which is a proper class,  (  o F R  |  `  ( A  X.  B
) ) can be a set by ofmresex 6052, allowing it to be used as a function or structure argument. By ofmresval 6051, the restricted operation map values are the same as the original values, allowing theorems for  o F R to be reused. (Contributed by NM, 20-Oct-2014.)
 |-  (  o F R  |`  ( A  X.  B ) )  =  (
 f  e.  A ,  g  e.  B  |->  ( f  o F R g ) )
 
Theoremofmresval 6051 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 (  o F R  |`  ( A  X.  B ) ) G )  =  ( F  o F R G ) )
 
Theoremofmresex 6052 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  (  o F R  |`  ( A  X.  B ) )  e.  _V )
 
Theoremsuppssof1 6053* 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  o F O B ) " ( _V  \  { Z }
 ) )  C_  L )
 
2.4.11  First and second members of an ordered pair
 
Syntaxc1st 6054 Extend the definition of a class to include the first member an ordered pair function.
 class  1st
 
Syntaxc2nd 6055 Extend the definition of a class to include the second member an ordered pair function.
 class  2nd
 
Definitiondf-1st 6056 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6062 proves that it does this. For example,  ( 1st ` 
<. 3 ,  4
>. )  =  3. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5141 and op1stb 4541). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 1st  =  ( x  e.  _V  |->  U. dom  {  x } )
 
Definitiondf-2nd 6057 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6063 proves that it does this. For example,  ( 2nd ` 
<. 3 ,  4
>. )  =  4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5144 and op2ndb 5143). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 2nd  =  ( x  e.  _V  |->  U. ran  {  x } )
 
Theorem1stval 6058 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.)
 |-  ( 1st `  A )  =  U. dom  {  A }
 
Theorem2ndval 6059 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.)
 |-  ( 2nd `  A )  =  U. ran  {  A }
 
Theorem1st0 6060 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 1st `  (/) )  =  (/)
 
Theorem2nd0 6061 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 2nd `  (/) )  =  (/)
 
Theoremop1st 6062 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 6063 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 2nd `  <. A ,  B >. )  =  B
 
Theoremop1std 6064 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( C  =  <. A ,  B >.  ->  ( 1st `  C )  =  A )
 
Theoremop2ndd 6065 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( C  =  <. A ,  B >.  ->  ( 2nd `  C )  =  B )
 
Theoremop1stg 6066 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
 
Theoremop2ndg 6067 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
 
Theoremot1stg 6068 Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6068, ot2ndg 6069, ot3rdg 6070.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
 
Theoremot2ndg 6069 Extract the second member of an ordered triple. (See ot1stg 6068 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
 
Theoremot3rdg 6070 Extract the third member of an ordered triple. (See ot1stg 6068 comment.) (Contributed by NM, 3-Apr-2015.)
 |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
 
Theorem1stval2 6071 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
 
Theorem2ndval2 6072 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^| |^| `' { A } )
 
Theoremfo1st 6073 The  1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 1st : _V -onto-> _V
 
Theoremfo2nd 6074 The  2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 2nd : _V -onto-> _V
 
Theoremf1stres 6075 Mapping of a restriction of the 
1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) --> A
 
Theoremf2ndres 6076 Mapping of a restriction of the 
2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) --> B
 
Theoremfo1stres 6077 Onto mapping of a restriction of the  1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
 |-  ( B  =/=  (/)  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
 
Theoremfo2ndres 6078 Onto mapping of a restriction of the  2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
 |-  ( A  =/=  (/)  ->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> B )
 
Theorem1st2val 6079* Value of an alternate definition of the  1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( { <. <. x ,  y >. ,  z >.  |  z  =  x } `  A )  =  ( 1st `  A )
 
Theorem2nd2val 6080* Value of an alternate definition of the  2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  ( { <. <. x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A )
 
Theorem1stcof 6081 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
 |-  ( F : A --> ( B  X.  C ) 
 ->  ( 1st  o.  F ) : A --> B )
 
Theorem2ndcof 6082 Composition of the first member function with another function. (Contributed by FL, 15-Oct-2012.)
 |-  ( F : A --> ( B  X.  C ) 
 ->  ( 2nd  o.  F ) : A --> C )
 
Theoremxp1st 6083 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  ( B  X.  C )  ->  ( 1st `  A )  e.  B )
 
Theoremxp2nd 6084 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  ( B  X.  C )  ->  ( 2nd `  A )  e.  C )
 
Theoremelxp6 6085 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5147. (Contributed by NM, 9-Oct-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >.  /\  (
 ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
 
Theoremelxp7 6086 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5147. (Contributed by NM, 19-Aug-2006.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
 
Theoremdifxp 6087 Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)
 |-  ( ( C  X.  D )  \  ( A  X.  B ) )  =  ( ( ( C  \  A )  X.  D )  u.  ( C  X.  ( D  \  B ) ) )
 
Theoremdifxp1 6088 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
 |-  ( ( A  \  B )  X.  C )  =  ( ( A  X.  C )  \  ( B  X.  C ) )
 
Theoremdifxp2 6089 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
 |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )
 
Theoremeqopi 6090 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  ( V  X.  W ) 
 /\  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C )
 )  ->  A  =  <. B ,  C >. )
 
Theoremxp2 6091* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  X.  B )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x )  e.  B ) }
 
Theoremunielxp 6092 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
 
Theorem1st2nd2 6093 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
 |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theorem1st2ndb 6094 Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)
 |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theoremxpopth 6095 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
 |-  ( ( A  e.  ( C  X.  D ) 
 /\  B  e.  ( R  X.  S ) ) 
 ->  ( ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B ) )  <->  A  =  B ) )
 
Theoremeqop 6096 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C )
 ) )
 
Theoremeqop2 6097 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =  <. B ,  C >.  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) ) )
 
Theoremop1steq 6098* Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
 |-  ( A  e.  ( V  X.  W )  ->  ( ( 1st `  A )  =  B  <->  E. x  A  =  <. B ,  x >. ) )
 
Theorem2nd1st 6099 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
 |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A ) >. )
 
Theorem1st2nd 6100 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
 |-  ( ( Rel  B  /\  A  e.  B ) 
 ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
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