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Theorem resfval2 15309
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 4720 . . . 4  |-  <. F ,  G >.  e.  _V
21a1i 11 . . 3  |-  ( ph  -> 
<. F ,  G >.  e. 
_V )
3 resfval.d . . 3  |-  ( ph  ->  H  e.  W )
42, 3resfval 15308 . 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 6811 . . . . 5  |-  ( ( F  e.  V  /\  G  e.  X )  ->  ( 1st `  <. F ,  G >. )  =  F )
85, 6, 7syl2anc 661 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
9 resfval2.d . . . . . . 7  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
10 fndm 5686 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
119, 10syl 16 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1211dmeqd 5215 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
13 dmxpid 5232 . . . . 5  |-  dom  ( S  X.  S )  =  S
1412, 13syl6eq 2514 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
158, 14reseq12d 5284 . . 3  |-  ( ph  ->  ( ( 1st `  <. F ,  G >. )  |` 
dom  dom  H )  =  ( F  |`  S ) )
16 op2ndg 6812 . . . . . . . 8  |-  ( ( F  e.  V  /\  G  e.  X )  ->  ( 2nd `  <. F ,  G >. )  =  G )
175, 6, 16syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
1817fveq1d 5874 . . . . . 6  |-  ( ph  ->  ( ( 2nd `  <. F ,  G >. ) `  z )  =  ( G `  z ) )
1918reseq1d 5282 . . . . 5  |-  ( ph  ->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) )  =  ( ( G `  z
)  |`  ( H `  z ) ) )
2011, 19mpteq12dv 4535 . . . 4  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) ) )  =  ( z  e.  ( S  X.  S ) 
|->  ( ( G `  z )  |`  ( H `  z )
) ) )
21 fveq2 5872 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
22 df-ov 6299 . . . . . . 7  |-  ( x G y )  =  ( G `  <. x ,  y >. )
2321, 22syl6eqr 2516 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( x G y ) )
24 fveq2 5872 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
25 df-ov 6299 . . . . . . 7  |-  ( x H y )  =  ( H `  <. x ,  y >. )
2624, 25syl6eqr 2516 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
2723, 26reseq12d 5284 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( G `
 z )  |`  ( H `  z ) )  =  ( ( x G y )  |`  ( x H y ) ) )
2827mpt2mpt 6393 . . . 4  |-  ( z  e.  ( S  X.  S )  |->  ( ( G `  z )  |`  ( H `  z
) ) )  =  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  (
x H y ) ) )
2920, 28syl6eq 2514 . . 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 4227 . 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 2498 1  |-  ( ph  ->  ( <. F ,  G >. 
|`f 
H )  =  <. ( F  |`  S ) ,  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  ( x H y ) ) ) >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038    |-> cmpt 4515    X. cxp 5006   dom cdm 5008    |` cres 5010    Fn wfn 5589   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798    |`f cresf 15273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-resf 15277
This theorem is referenced by:  funcrngcsetc  32950  funcringcsetc  32987
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