MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  offres Unicode version

Theorem offres 6286
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 3530 . . . . . 6  |-  ( ( dom  F  i^i  dom  G )  i^i  D ) 
C_  D
21sseli 3312 . . . . 5  |-  ( x  e.  ( ( dom 
F  i^i  dom  G )  i^i  D )  ->  x  e.  D )
3 fvres 5712 . . . . . 6  |-  ( x  e.  D  ->  (
( F  |`  D ) `
 x )  =  ( F `  x
) )
4 fvres 5712 . . . . . 6  |-  ( x  e.  D  ->  (
( G  |`  D ) `
 x )  =  ( G `  x
) )
53, 4oveq12d 6066 . . . . 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 4259 . . 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 3526 . . . . 5  |-  ( D  i^i  ( dom  F  i^i  dom  G ) )  =  ( ( D  i^i  dom  F )  i^i  ( D  i^i  dom  G ) )
9 incom 3501 . . . . 5  |-  ( ( dom  F  i^i  dom  G )  i^i  D )  =  ( D  i^i  ( dom  F  i^i  dom  G ) )
10 dmres 5134 . . . . . 6  |-  dom  ( F  |`  D )  =  ( D  i^i  dom  F )
11 dmres 5134 . . . . . 6  |-  dom  ( G  |`  D )  =  ( D  i^i  dom  G )
1210, 11ineq12i 3508 . . . . 5  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( D  i^i  dom 
F )  i^i  ( D  i^i  dom  G )
)
138, 9, 123eqtr4ri 2443 . . . 4  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( dom  F  i^i  dom  G )  i^i 
D )
14 eqid 2412 . . . 4  |-  ( ( ( F  |`  D ) `
 x ) R ( ( G  |`  D ) `  x
) )  =  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )
1513, 14mpteq12i 4261 . . 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 5159 . . 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 2443 . 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 6285 . . 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 5112 . 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 5152 . . 3  |-  ( F  e.  V  ->  ( F  |`  D )  e. 
_V )
21 resexg 5152 . . 3  |-  ( G  e.  W  ->  ( G  |`  D )  e. 
_V )
22 offval3 6285 . . 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 464 . 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 2470 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    i^i cin 3287    e. cmpt 4234   dom cdm 4845    |` cres 4847   ` cfv 5421  (class class class)co 6048    o Fcof 6270
This theorem is referenced by:  tsmsadd  18137  jensen  20788  pwssplit2  27065  pwssplit3  27066  islindf4  27184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272
  Copyright terms: Public domain W3C validator