MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-of Unicode version

Definition df-of 6040
Description: 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.)
Assertion
Ref Expression
df-of  |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
Distinct variable group:    f, g, x, R

Detailed syntax breakdown of Definition df-of
StepHypRef Expression
1 cR . . 3  class  R
21cof 6038 . 2  class  o F R
3 vf . . 3  set  f
4 vg . . 3  set  g
5 cvv 2790 . . 3  class  _V
6 vx . . . 4  set  x
73cv 1623 . . . . . 6  class  f
87cdm 4689 . . . . 5  class  dom  f
94cv 1623 . . . . . 6  class  g
109cdm 4689 . . . . 5  class  dom  g
118, 10cin 3153 . . . 4  class  ( dom  f  i^i  dom  g
)
126cv 1623 . . . . . 6  class  x
1312, 7cfv 5222 . . . . 5  class  ( f `
 x )
1412, 9cfv 5222 . . . . 5  class  ( g `
 x )
1513, 14, 1co 5820 . . . 4  class  ( ( f `  x ) R ( g `  x ) )
166, 11, 15cmpt 4079 . . 3  class  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) )
173, 4, 5, 5, 16cmpt2 5822 . 2  class  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
182, 17wceq 1624 1  wff  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  ofeq  6042  ofexg  6044  nfof  6045  offval  6047  offval3  6053  ofmres  6078
  Copyright terms: Public domain W3C validator