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Theorem resfval 14094
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 14063 . . 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 734 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
f  =  F )
43fveq2d 5735 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
5 simprr 735 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  h  =  H )
65dmeqd 5075 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  dom  h  =  dom  H
)
76dmeqd 5075 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  ->  dom  dom  h  =  dom  dom 
H )
84, 7reseq12d 5150 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( 1st `  f
)  |`  dom  dom  h
)  =  ( ( 1st `  F )  |`  dom  dom  H )
)
93fveq2d 5735 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
109fveq1d 5733 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( 2nd `  f
) `  x )  =  ( ( 2nd `  F ) `  x
) )
115fveq1d 5733 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( h `  x
)  =  ( H `
 x ) )
1210, 11reseq12d 5150 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  h  =  H ) )  -> 
( ( ( 2nd `  f ) `  x
)  |`  ( h `  x ) )  =  ( ( ( 2nd `  F ) `  x
)  |`  ( H `  x ) ) )
136, 12mpteq12dv 4290 . . 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 3994 . 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 2966 . . 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 2966 . . 3  |-  ( H  e.  W  ->  H  e.  _V )
2018, 19syl 16 . 2  |-  ( ph  ->  H  e.  _V )
21 opex 4430 . . 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 6204 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    e. cmpt 4269   dom cdm 4881    |` cres 4883   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351    |`f cresf 14059
This theorem is referenced by:  resfval2  14095  resf1st  14096  resf2nd  14097  funcres  14098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-resf 14063
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