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Theorem resfval2 14082
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 4419 . . . 4  |-  <. F ,  G >.  e.  _V
21a1i 11 . . 3  |-  ( ph  -> 
<. F ,  G >.  e. 
_V )
3 resfval.d . . 3  |-  ( ph  ->  H  e.  W )
42, 3resfval 14081 . 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 6351 . . . . 5  |-  ( ( F  e.  V  /\  G  e.  X )  ->  ( 1st `  <. F ,  G >. )  =  F )
85, 6, 7syl2anc 643 . . . 4  |-  ( ph  ->  ( 1st `  <. F ,  G >. )  =  F )
9 resfval2.d . . . . . . 7  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
10 fndm 5536 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
119, 10syl 16 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1211dmeqd 5064 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
13 dmxpid 5081 . . . . 5  |-  dom  ( S  X.  S )  =  S
1412, 13syl6eq 2483 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
158, 14reseq12d 5139 . . 3  |-  ( ph  ->  ( ( 1st `  <. F ,  G >. )  |` 
dom  dom  H )  =  ( F  |`  S ) )
16 op2ndg 6352 . . . . . . . 8  |-  ( ( F  e.  V  /\  G  e.  X )  ->  ( 2nd `  <. F ,  G >. )  =  G )
175, 6, 16syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
1817fveq1d 5722 . . . . . 6  |-  ( ph  ->  ( ( 2nd `  <. F ,  G >. ) `  z )  =  ( G `  z ) )
1918reseq1d 5137 . . . . 5  |-  ( ph  ->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) )  =  ( ( G `  z
)  |`  ( H `  z ) ) )
2011, 19mpteq12dv 4279 . . . 4  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) ) )  =  ( z  e.  ( S  X.  S ) 
|->  ( ( G `  z )  |`  ( H `  z )
) ) )
21 fveq2 5720 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
22 df-ov 6076 . . . . . . 7  |-  ( x G y )  =  ( G `  <. x ,  y >. )
2321, 22syl6eqr 2485 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( x G y ) )
24 fveq2 5720 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
25 df-ov 6076 . . . . . . 7  |-  ( x H y )  =  ( H `  <. x ,  y >. )
2624, 25syl6eqr 2485 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
2723, 26reseq12d 5139 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( G `
 z )  |`  ( H `  z ) )  =  ( ( x G y )  |`  ( x H y ) ) )
2827mpt2mpt 6157 . . . 4  |-  ( z  e.  ( S  X.  S )  |->  ( ( G `  z )  |`  ( H `  z
) ) )  =  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  (
x H y ) ) )
2920, 28syl6eq 2483 . . 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 3984 . 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 2467 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 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809    e. cmpt 4258    X. cxp 4868   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340    |`f cresf 14046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-resf 14050
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