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Theorem resfval2 13783
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 4253 . . . 4  |-  <. F ,  G >.  e.  _V
21a1i 10 . . 3  |-  ( ph  -> 
<. F ,  G >.  e. 
_V )
3 resfval.d . . 3  |-  ( ph  ->  H  e.  W )
42, 3resfval 13782 . 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 6148 . . . . 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 5359 . . . . . . 7  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
119, 10syl 15 . . . . . 6  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1211dmeqd 4897 . . . . 5  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
13 dmxpid 4914 . . . . 5  |-  dom  ( S  X.  S )  =  S
1412, 13syl6eq 2344 . . . 4  |-  ( ph  ->  dom  dom  H  =  S )
158, 14reseq12d 4972 . . 3  |-  ( ph  ->  ( ( 1st `  <. F ,  G >. )  |` 
dom  dom  H )  =  ( F  |`  S ) )
16 op2ndg 6149 . . . . . . . 8  |-  ( ( F  e.  V  /\  G  e.  X )  ->  ( 2nd `  <. F ,  G >. )  =  G )
175, 6, 16syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. F ,  G >. )  =  G )
1817fveq1d 5543 . . . . . 6  |-  ( ph  ->  ( ( 2nd `  <. F ,  G >. ) `  z )  =  ( G `  z ) )
1918reseq1d 4970 . . . . 5  |-  ( ph  ->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) )  =  ( ( G `  z
)  |`  ( H `  z ) ) )
2011, 19mpteq12dv 4114 . . . 4  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  <. F ,  G >. ) `  z )  |`  ( H `  z
) ) )  =  ( z  e.  ( S  X.  S ) 
|->  ( ( G `  z )  |`  ( H `  z )
) ) )
21 fveq2 5541 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
22 df-ov 5877 . . . . . . 7  |-  ( x G y )  =  ( G `  <. x ,  y >. )
2321, 22syl6eqr 2346 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( x G y ) )
24 fveq2 5541 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
25 df-ov 5877 . . . . . . 7  |-  ( x H y )  =  ( H `  <. x ,  y >. )
2624, 25syl6eqr 2346 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
2723, 26reseq12d 4972 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( G `
 z )  |`  ( H `  z ) )  =  ( ( x G y )  |`  ( x H y ) ) )
2827mpt2mpt 5955 . . . 4  |-  ( z  e.  ( S  X.  S )  |->  ( ( G `  z )  |`  ( H `  z
) ) )  =  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  (
x H y ) ) )
2920, 28syl6eq 2344 . . 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 3820 . 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 2328 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093    X. cxp 4703   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137    |`f cresf 13747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-resf 13751
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