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Theorem offres 5999
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 3265 . . . . . 6  |-  ( ( dom  F  i^i  dom  G )  i^i  D ) 
C_  D
21sseli 3061 . . . . 5  |-  ( x  e.  ( ( dom 
F  i^i  dom  G )  i^i  D )  ->  x  e.  D )
3 fvres 5411 . . . . . 6  |-  ( x  e.  D  ->  (
( F  |`  D ) `
 x )  =  ( F `  x
) )
4 fvres 5411 . . . . . 6  |-  ( x  e.  D  ->  (
( G  |`  D ) `
 x )  =  ( G `  x
) )
53, 4oveq12d 5758 . . . . 5  |-  ( x  e.  D  ->  (
( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
62, 5syl 14 . . . 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 3982 . . 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 3261 . . . . 5  |-  ( D  i^i  ( dom  F  i^i  dom  G ) )  =  ( ( D  i^i  dom  F )  i^i  ( D  i^i  dom  G ) )
9 incom 3236 . . . . 5  |-  ( ( dom  F  i^i  dom  G )  i^i  D )  =  ( D  i^i  ( dom  F  i^i  dom  G ) )
10 dmres 4808 . . . . . 6  |-  dom  ( F  |`  D )  =  ( D  i^i  dom  F )
11 dmres 4808 . . . . . 6  |-  dom  ( G  |`  D )  =  ( D  i^i  dom  G )
1210, 11ineq12i 3243 . . . . 5  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( D  i^i  dom 
F )  i^i  ( D  i^i  dom  G )
)
138, 9, 123eqtr4ri 2147 . . . 4  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( dom  F  i^i  dom  G )  i^i 
D )
14 eqid 2115 . . . 4  |-  ( ( ( F  |`  D ) `
 x ) R ( ( G  |`  D ) `  x
) )  =  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )
1513, 14mpteq12i 3984 . . 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 4836 . . 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 2147 . 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 5998 . . 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 4786 . 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 4827 . . 3  |-  ( F  e.  V  ->  ( F  |`  D )  e. 
_V )
21 resexg 4827 . . 3  |-  ( G  e.  W  ->  ( G  |`  D )  e. 
_V )
22 offval3 5998 . . 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 285 . 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 2174 1  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  oF R G )  |`  D )  =  ( ( F  |`  D )  oF R ( G  |`  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   _Vcvv 2658    i^i cin 3038    |-> cmpt 3957   dom cdm 4507    |` cres 4509   ` cfv 5091  (class class class)co 5740    oFcof 5946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-of 5948
This theorem is referenced by: (None)
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