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Theorem offres 6092
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  o F R G )  |`  D )  =  ( ( F  |`  D )  o F R ( G  |`  D )
) )

Proof of Theorem offres
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 inss2 3390 . . . . . 6  |-  ( ( dom  F  i^i  dom  G )  i^i  D ) 
C_  D
21sseli 3176 . . . . 5  |-  ( x  e.  ( ( dom 
F  i^i  dom  G )  i^i  D )  ->  x  e.  D )
3 fvres 5542 . . . . . 6  |-  ( x  e.  D  ->  (
( F  |`  D ) `
 x )  =  ( F `  x
) )
4 fvres 5542 . . . . . 6  |-  ( x  e.  D  ->  (
( G  |`  D ) `
 x )  =  ( G `  x
) )
53, 4oveq12d 5876 . . . . 5  |-  ( x  e.  D  ->  (
( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
62, 5syl 15 . . . 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 4102 . . 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 3386 . . . . 5  |-  ( D  i^i  ( dom  F  i^i  dom  G ) )  =  ( ( D  i^i  dom  F )  i^i  ( D  i^i  dom  G ) )
9 incom 3361 . . . . 5  |-  ( ( dom  F  i^i  dom  G )  i^i  D )  =  ( D  i^i  ( dom  F  i^i  dom  G ) )
10 dmres 4976 . . . . . 6  |-  dom  ( F  |`  D )  =  ( D  i^i  dom  F )
11 dmres 4976 . . . . . 6  |-  dom  ( G  |`  D )  =  ( D  i^i  dom  G )
1210, 11ineq12i 3368 . . . . 5  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( D  i^i  dom 
F )  i^i  ( D  i^i  dom  G )
)
138, 9, 123eqtr4ri 2314 . . . 4  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( dom  F  i^i  dom  G )  i^i 
D )
14 eqid 2283 . . . 4  |-  ( ( ( F  |`  D ) `
 x ) R ( ( G  |`  D ) `  x
) )  =  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )
1513, 14mpteq12i 4104 . . 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 5001 . . 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 2314 . 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 6091 . . 3  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  o F R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
1918reseq1d 4954 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  o F R G )  |`  D )  =  ( ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  |`  D ) )
20 resexg 4994 . . 3  |-  ( F  e.  V  ->  ( F  |`  D )  e. 
_V )
21 resexg 4994 . . 3  |-  ( G  e.  W  ->  ( G  |`  D )  e. 
_V )
22 offval3 6091 . . 3  |-  ( ( ( F  |`  D )  e.  _V  /\  ( G  |`  D )  e. 
_V )  ->  (
( F  |`  D )  o F R ( G  |`  D )
)  =  ( x  e.  ( dom  ( F  |`  D )  i^i 
dom  ( G  |`  D ) )  |->  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) ) ) )
2320, 21, 22syl2an 463 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  |`  D )  o F R ( G  |`  D ) )  =  ( x  e.  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  |->  ( ( ( F  |`  D ) `  x ) R ( ( G  |`  D ) `
 x ) ) ) )
2417, 19, 233eqtr4a 2341 1  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  o F R G )  |`  D )  =  ( ( F  |`  D )  o F R ( G  |`  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    e. cmpt 4077   dom cdm 4689    |` cres 4691   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  tsmsadd  17829  jensen  20283  pwssplit2  27189  pwssplit3  27190  islindf4  27308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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