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Definition df-resf 13751
Description: Define the restriction of a functor to a subcategory (analogue of df-res 4717). (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
df-resf  |-  |`f  =  ( f  e. 
_V ,  h  e. 
_V  |->  <. ( ( 1st `  f )  |`  dom  dom  h ) ,  ( x  e.  dom  h  |->  ( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) ) )
>. )
Distinct variable group:    f, h, x

Detailed syntax breakdown of Definition df-resf
StepHypRef Expression
1 cresf 13747 . 2  class  |`f
2 vf . . 3  set  f
3 vh . . 3  set  h
4 cvv 2801 . . 3  class  _V
52cv 1631 . . . . . 6  class  f
6 c1st 6136 . . . . . 6  class  1st
75, 6cfv 5271 . . . . 5  class  ( 1st `  f )
83cv 1631 . . . . . . 7  class  h
98cdm 4705 . . . . . 6  class  dom  h
109cdm 4705 . . . . 5  class  dom  dom  h
117, 10cres 4707 . . . 4  class  ( ( 1st `  f )  |`  dom  dom  h )
12 vx . . . . 5  set  x
1312cv 1631 . . . . . . 7  class  x
14 c2nd 6137 . . . . . . . 8  class  2nd
155, 14cfv 5271 . . . . . . 7  class  ( 2nd `  f )
1613, 15cfv 5271 . . . . . 6  class  ( ( 2nd `  f ) `
 x )
1713, 8cfv 5271 . . . . . 6  class  ( h `
 x )
1816, 17cres 4707 . . . . 5  class  ( ( ( 2nd `  f
) `  x )  |`  ( h `  x
) )
1912, 9, 18cmpt 4093 . . . 4  class  ( x  e.  dom  h  |->  ( ( ( 2nd `  f
) `  x )  |`  ( h `  x
) ) )
2011, 19cop 3656 . . 3  class  <. (
( 1st `  f
)  |`  dom  dom  h
) ,  ( x  e.  dom  h  |->  ( ( ( 2nd `  f
) `  x )  |`  ( h `  x
) ) ) >.
212, 3, 4, 4, 20cmpt2 5876 . 2  class  ( f  e.  _V ,  h  e.  _V  |->  <. ( ( 1st `  f )  |`  dom  dom  h ) ,  ( x  e.  dom  h  |->  ( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) ) )
>. )
221, 21wceq 1632 1  wff  |`f  =  ( f  e. 
_V ,  h  e. 
_V  |->  <. ( ( 1st `  f )  |`  dom  dom  h ) ,  ( x  e.  dom  h  |->  ( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) ) )
>. )
Colors of variables: wff set class
This definition is referenced by:  resfval  13782
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