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Theorem resfval 13766
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 13735 . . 3  |-  |`f  =  ( f  e. 
_V ,  h  e. 
_V  |->  <. ( ( 1st `  f )  |`  dom  dom  h ) ,  ( x  e.  dom  h  |->  ( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) ) )
>. )
21a1i 10 . 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 732 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
f  =  F )
43fveq2d 5529 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
5 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  h  =  H )
65dmeqd 4881 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  dom  h  =  dom  H
)
76dmeqd 4881 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  dom  dom  h  =  dom  dom 
H )
84, 7reseq12d 4956 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( 1st `  f
)  |`  dom  dom  h
)  =  ( ( 1st `  F )  |`  dom  dom  H )
)
93fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
109fveq1d 5527 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( 2nd `  f
) `  x )  =  ( ( 2nd `  F ) `  x
) )
115fveq1d 5527 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( h `  x
)  =  ( H `
 x ) )
1210, 11reseq12d 4956 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) )  =  ( ( ( 2nd `  F ) `  x
)  |`  ( H `  x ) ) )
136, 12mpteq12dv 4098 . . 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 3804 . 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 2796 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
1715, 16syl 15 . 2  |-  ( ph  ->  F  e.  _V )
18 resfval.d . . 3  |-  ( ph  ->  H  e.  W )
19 elex 2796 . . 3  |-  ( H  e.  W  ->  H  e.  _V )
2018, 19syl 15 . 2  |-  ( ph  ->  H  e.  _V )
21 opex 4237 . . 3  |-  <. (
( 1st `  F
)  |`  dom  dom  H
) ,  ( x  e.  dom  H  |->  ( ( ( 2nd `  F
) `  x )  |`  ( H `  x
) ) ) >.  e.  _V
2221a1i 10 . 2  |-  ( ph  -> 
<. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( x  e.  dom  H  |->  ( ( ( 2nd `  F
) `  x )  |`  ( H `  x
) ) ) >.  e.  _V )
232, 14, 17, 20, 22ovmpt2d 5975 1  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( x  e.  dom  H  |->  ( ( ( 2nd `  F
) `  x )  |`  ( H `  x
) ) ) >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077   dom cdm 4689    |` cres 4691   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121    |`f cresf 13731
This theorem is referenced by:  resfval2  13767  resf1st  13768  resf2nd  13769  funcres  13770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-resf 13735
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