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Theorem resfval2 13767
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 )
resfval2.g  |-  ( ph  ->  G  e.  X )
resfval2.d  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
Assertion
Ref Expression
resfval2  |-  ( ph  ->  ( <. F ,  G >. 
|`f 
H )  =  <. ( F  |`  S ) ,  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  ( x H y ) ) ) >.
)
Distinct variable groups:    x, F    x, y, G    x, H, y    ph, x    x, S, y
Allowed substitution hints:    ph( y)    F( y)    V( x, y)    W( x, y)    X( x, y)

Proof of Theorem resfval2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 opex 4237 . . . 4  |-  <. F ,  G >.  e.  _V
21a1i 10 . . 3  |-  ( ph  -> 
<. F ,  G >.  e. 
_V )
3 resfval.d . . 3  |-  ( ph  ->  H  e.  W )
42, 3resfval 13766 . 2  |-  ( ph  ->  ( <. F ,  G >. 
|`f 
H )  =  <. ( ( 1st `  <. F ,  G >. )  |` 
dom  dom  H ) ,  ( z  e.  dom  H 
|->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) ) ) >.
)
5 resfval.c . . . . 5  |-  ( ph  ->  F  e.  V )
6 resfval2.g . . . . 5  |-  ( ph  ->  G  e.  X )
7 op1stg 6132 . . . . 5  |-  ( ( F  e.  V  /\  G  e.  X )  ->  ( 1st `  <. F ,  G >. )  =  F )
85, 6, 7syl2anc 642 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
9 resfval2.d . . . . . . 7  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
10 fndm 5343 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
119, 10syl 15 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1211dmeqd 4881 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
13 dmxpid 4898 . . . . 5  |-  dom  ( S  X.  S )  =  S
1412, 13syl6eq 2331 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
158, 14reseq12d 4956 . . 3  |-  ( ph  ->  ( ( 1st `  <. F ,  G >. )  |` 
dom  dom  H )  =  ( F  |`  S ) )
16 op2ndg 6133 . . . . . . . 8  |-  ( ( F  e.  V  /\  G  e.  X )  ->  ( 2nd `  <. F ,  G >. )  =  G )
175, 6, 16syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
1817fveq1d 5527 . . . . . 6  |-  ( ph  ->  ( ( 2nd `  <. F ,  G >. ) `  z )  =  ( G `  z ) )
1918reseq1d 4954 . . . . 5  |-  ( ph  ->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) )  =  ( ( G `  z
)  |`  ( H `  z ) ) )
2011, 19mpteq12dv 4098 . . . 4  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) ) )  =  ( z  e.  ( S  X.  S ) 
|->  ( ( G `  z )  |`  ( H `  z )
) ) )
21 fveq2 5525 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
22 df-ov 5861 . . . . . . 7  |-  ( x G y )  =  ( G `  <. x ,  y >. )
2321, 22syl6eqr 2333 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( x G y ) )
24 fveq2 5525 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
25 df-ov 5861 . . . . . . 7  |-  ( x H y )  =  ( H `  <. x ,  y >. )
2624, 25syl6eqr 2333 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
2723, 26reseq12d 4956 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( G `
 z )  |`  ( H `  z ) )  =  ( ( x G y )  |`  ( x H y ) ) )
2827mpt2mpt 5939 . . . 4  |-  ( z  e.  ( S  X.  S )  |->  ( ( G `  z )  |`  ( H `  z
) ) )  =  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  (
x H y ) ) )
2920, 28syl6eq 2331 . . 3  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) ) )  =  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  (
x H y ) ) ) )
3015, 29opeq12d 3804 . 2  |-  ( ph  -> 
<. ( ( 1st `  <. F ,  G >. )  |` 
dom  dom  H ) ,  ( z  e.  dom  H 
|->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) ) ) >.  =  <. ( F  |`  S ) ,  ( x  e.  S , 
y  e.  S  |->  ( ( x G y )  |`  ( x H y ) ) ) >. )
314, 30eqtrd 2315 1  |-  ( ph  ->  ( <. F ,  G >. 
|`f 
H )  =  <. ( F  |`  S ) ,  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  ( x H y ) ) ) >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077    X. cxp 4687   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121    |`f cresf 13731
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-csb 3082  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-iun 3907  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-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-resf 13735
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