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Theorem resfval 15136
Description: Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resfval.c  |-  ( ph  ->  F  e.  V )
resfval.d  |-  ( ph  ->  H  e.  W )
Assertion
Ref Expression
resfval  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( x  e.  dom  H  |->  ( ( ( 2nd `  F
) `  x )  |`  ( H `  x
) ) ) >.
)
Distinct variable groups:    x, F    x, H    ph, x
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem resfval
Dummy variables  f  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-resf 15105 . . 3  |-  |`f  =  ( f  e. 
_V ,  h  e. 
_V  |->  <. ( ( 1st `  f )  |`  dom  dom  h ) ,  ( x  e.  dom  h  |->  ( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) ) )
>. )
21a1i 11 . 2  |-  ( ph  -> 
|`f 
=  ( f  e. 
_V ,  h  e. 
_V  |->  <. ( ( 1st `  f )  |`  dom  dom  h ) ,  ( x  e.  dom  h  |->  ( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) ) )
>. ) )
3 simprl 755 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
f  =  F )
43fveq2d 5876 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
5 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  h  =  H )
65dmeqd 5211 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  dom  h  =  dom  H
)
76dmeqd 5211 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  dom  dom  h  =  dom  dom 
H )
84, 7reseq12d 5280 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( 1st `  f
)  |`  dom  dom  h
)  =  ( ( 1st `  F )  |`  dom  dom  H )
)
93fveq2d 5876 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
109fveq1d 5874 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( 2nd `  f
) `  x )  =  ( ( 2nd `  F ) `  x
) )
115fveq1d 5874 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( h `  x
)  =  ( H `
 x ) )
1210, 11reseq12d 5280 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) )  =  ( ( ( 2nd `  F ) `  x
)  |`  ( H `  x ) ) )
136, 12mpteq12dv 4531 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( x  e.  dom  h  |->  ( ( ( 2nd `  f ) `
 x )  |`  ( h `  x
) ) )  =  ( x  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  x
)  |`  ( H `  x ) ) ) )
148, 13opeq12d 4227 . 2  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  <. ( ( 1st `  f
)  |`  dom  dom  h
) ,  ( x  e.  dom  h  |->  ( ( ( 2nd `  f
) `  x )  |`  ( h `  x
) ) ) >.  =  <. ( ( 1st `  F )  |`  dom  dom  H ) ,  ( x  e.  dom  H  |->  ( ( ( 2nd `  F
) `  x )  |`  ( H `  x
) ) ) >.
)
15 resfval.c . . 3  |-  ( ph  ->  F  e.  V )
16 elex 3127 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
1715, 16syl 16 . 2  |-  ( ph  ->  F  e.  _V )
18 resfval.d . . 3  |-  ( ph  ->  H  e.  W )
19 elex 3127 . . 3  |-  ( H  e.  W  ->  H  e.  _V )
2018, 19syl 16 . 2  |-  ( ph  ->  H  e.  _V )
21 opex 4717 . . 3  |-  <. (
( 1st `  F
)  |`  dom  dom  H
) ,  ( x  e.  dom  H  |->  ( ( ( 2nd `  F
) `  x )  |`  ( H `  x
) ) ) >.  e.  _V
2221a1i 11 . 2  |-  ( ph  -> 
<. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( x  e.  dom  H  |->  ( ( ( 2nd `  F
) `  x )  |`  ( H `  x
) ) ) >.  e.  _V )
232, 14, 17, 20, 22ovmpt2d 6425 1  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( x  e.  dom  H  |->  ( ( ( 2nd `  F
) `  x )  |`  ( H `  x
) ) ) >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   <.cop 4039    |-> cmpt 4511   dom cdm 5005    |` cres 5007   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794    |`f cresf 15101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-resf 15105
This theorem is referenced by:  resfval2  15137  resf1st  15138  resf2nd  15139  funcres  15140
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