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Theorem resfval2 14824
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 4577 . . . 4  |-  <. F ,  G >.  e.  _V
21a1i 11 . . 3  |-  ( ph  -> 
<. F ,  G >.  e. 
_V )
3 resfval.d . . 3  |-  ( ph  ->  H  e.  W )
42, 3resfval 14823 . 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 6610 . . . . 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 5531 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
119, 10syl 16 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1211dmeqd 5063 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
13 dmxpid 5080 . . . . 5  |-  dom  ( S  X.  S )  =  S
1412, 13syl6eq 2491 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
158, 14reseq12d 5132 . . 3  |-  ( ph  ->  ( ( 1st `  <. F ,  G >. )  |` 
dom  dom  H )  =  ( F  |`  S ) )
16 op2ndg 6611 . . . . . . . 8  |-  ( ( F  e.  V  /\  G  e.  X )  ->  ( 2nd `  <. F ,  G >. )  =  G )
175, 6, 16syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
1817fveq1d 5714 . . . . . 6  |-  ( ph  ->  ( ( 2nd `  <. F ,  G >. ) `  z )  =  ( G `  z ) )
1918reseq1d 5130 . . . . 5  |-  ( ph  ->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) )  =  ( ( G `  z
)  |`  ( H `  z ) ) )
2011, 19mpteq12dv 4391 . . . 4  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) ) )  =  ( z  e.  ( S  X.  S ) 
|->  ( ( G `  z )  |`  ( H `  z )
) ) )
21 fveq2 5712 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
22 df-ov 6115 . . . . . . 7  |-  ( x G y )  =  ( G `  <. x ,  y >. )
2321, 22syl6eqr 2493 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( x G y ) )
24 fveq2 5712 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
25 df-ov 6115 . . . . . . 7  |-  ( x H y )  =  ( H `  <. x ,  y >. )
2624, 25syl6eqr 2493 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
2723, 26reseq12d 5132 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( G `
 z )  |`  ( H `  z ) )  =  ( ( x G y )  |`  ( x H y ) ) )
2827mpt2mpt 6203 . . . 4  |-  ( z  e.  ( S  X.  S )  |->  ( ( G `  z )  |`  ( H `  z
) ) )  =  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  (
x H y ) ) )
2920, 28syl6eq 2491 . . 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 4088 . 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 2475 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 1369    e. wcel 1756   _Vcvv 2993   <.cop 3904    e. cmpt 4371    X. cxp 4859   dom cdm 4861    |` cres 4863    Fn wfn 5434   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   1stc1st 6596   2ndc2nd 6597    |`f cresf 14788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-iota 5402  df-fun 5441  df-fn 5442  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-resf 14792
This theorem is referenced by: (None)
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