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Theorem resfval 14052
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 14021 . . 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 733 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
f  =  F )
43fveq2d 5699 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
5 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  h  =  H )
65dmeqd 5039 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  dom  h  =  dom  H
)
76dmeqd 5039 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  dom  dom  h  =  dom  dom 
H )
84, 7reseq12d 5114 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( 1st `  f
)  |`  dom  dom  h
)  =  ( ( 1st `  F )  |`  dom  dom  H )
)
93fveq2d 5699 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
109fveq1d 5697 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( 2nd `  f
) `  x )  =  ( ( 2nd `  F ) `  x
) )
115fveq1d 5697 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( h `  x
)  =  ( H `
 x ) )
1210, 11reseq12d 5114 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) )  =  ( ( ( 2nd `  F ) `  x
)  |`  ( H `  x ) ) )
136, 12mpteq12dv 4255 . . 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 3960 . 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 2932 . . 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 2932 . . 3  |-  ( H  e.  W  ->  H  e.  _V )
2018, 19syl 16 . 2  |-  ( ph  ->  H  e.  _V )
21 opex 4395 . . 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 6168 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785    e. cmpt 4234   dom cdm 4845    |` cres 4847   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1stc1st 6314   2ndc2nd 6315    |`f cresf 14017
This theorem is referenced by:  resfval2  14053  resf1st  14054  resf2nd  14055  funcres  14056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-resf 14021
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