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Theorem offres 6571
Description: Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
offres  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  oF R G )  |`  D )  =  ( ( F  |`  D )  oF R ( G  |`  D )
) )

Proof of Theorem offres
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 inss2 3570 . . . . . 6  |-  ( ( dom  F  i^i  dom  G )  i^i  D ) 
C_  D
21sseli 3351 . . . . 5  |-  ( x  e.  ( ( dom 
F  i^i  dom  G )  i^i  D )  ->  x  e.  D )
3 fvres 5703 . . . . . 6  |-  ( x  e.  D  ->  (
( F  |`  D ) `
 x )  =  ( F `  x
) )
4 fvres 5703 . . . . . 6  |-  ( x  e.  D  ->  (
( G  |`  D ) `
 x )  =  ( G `  x
) )
53, 4oveq12d 6108 . . . . 5  |-  ( x  e.  D  ->  (
( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
62, 5syl 16 . . . 4  |-  ( x  e.  ( ( dom 
F  i^i  dom  G )  i^i  D )  -> 
( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
76mpteq2ia 4373 . . 3  |-  ( x  e.  ( ( dom 
F  i^i  dom  G )  i^i  D )  |->  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) ) )  =  ( x  e.  ( ( dom  F  i^i  dom  G )  i^i 
D )  |->  ( ( F `  x ) R ( G `  x ) ) )
8 inindi 3566 . . . . 5  |-  ( D  i^i  ( dom  F  i^i  dom  G ) )  =  ( ( D  i^i  dom  F )  i^i  ( D  i^i  dom  G ) )
9 incom 3542 . . . . 5  |-  ( ( dom  F  i^i  dom  G )  i^i  D )  =  ( D  i^i  ( dom  F  i^i  dom  G ) )
10 dmres 5130 . . . . . 6  |-  dom  ( F  |`  D )  =  ( D  i^i  dom  F )
11 dmres 5130 . . . . . 6  |-  dom  ( G  |`  D )  =  ( D  i^i  dom  G )
1210, 11ineq12i 3549 . . . . 5  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( D  i^i  dom 
F )  i^i  ( D  i^i  dom  G )
)
138, 9, 123eqtr4ri 2473 . . . 4  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( dom  F  i^i  dom  G )  i^i 
D )
14 eqid 2442 . . . 4  |-  ( ( ( F  |`  D ) `
 x ) R ( ( G  |`  D ) `  x
) )  =  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )
1513, 14mpteq12i 4375 . . 3  |-  ( x  e.  ( dom  ( F  |`  D )  i^i 
dom  ( G  |`  D ) )  |->  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) ) )  =  ( x  e.  ( ( dom  F  i^i  dom  G )  i^i 
D )  |->  ( ( ( F  |`  D ) `
 x ) R ( ( G  |`  D ) `  x
) ) )
16 resmpt3 5156 . . 3  |-  ( ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  |`  D )  =  ( x  e.  ( ( dom  F  i^i  dom  G )  i^i 
D )  |->  ( ( F `  x ) R ( G `  x ) ) )
177, 15, 163eqtr4ri 2473 . 2  |-  ( ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  |`  D )  =  ( x  e.  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  |->  ( ( ( F  |`  D ) `  x ) R ( ( G  |`  D ) `
 x ) ) )
18 offval3 6570 . . 3  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
1918reseq1d 5108 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  oF R G )  |`  D )  =  ( ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  |`  D ) )
20 resexg 5148 . . 3  |-  ( F  e.  V  ->  ( F  |`  D )  e. 
_V )
21 resexg 5148 . . 3  |-  ( G  e.  W  ->  ( G  |`  D )  e. 
_V )
22 offval3 6570 . . 3  |-  ( ( ( F  |`  D )  e.  _V  /\  ( G  |`  D )  e. 
_V )  ->  (
( F  |`  D )  oF R ( G  |`  D )
)  =  ( x  e.  ( dom  ( F  |`  D )  i^i 
dom  ( G  |`  D ) )  |->  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) ) ) )
2320, 21, 22syl2an 477 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  |`  D )  oF R ( G  |`  D ) )  =  ( x  e.  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  |->  ( ( ( F  |`  D ) `  x ) R ( ( G  |`  D ) `
 x ) ) ) )
2417, 19, 233eqtr4a 2500 1  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  oF R G )  |`  D )  =  ( ( F  |`  D )  oF R ( G  |`  D )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971    i^i cin 3326    e. cmpt 4349   dom cdm 4839    |` cres 4841   ` cfv 5417  (class class class)co 6090    oFcof 6317
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-rep 4402  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319
This theorem is referenced by:  pwssplit2  17140  pwssplit3  17141  islindf4  18266  tsmsadd  19720  jensen  22381
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