ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  offres Unicode version

Theorem offres 6138
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 3358 . . . . . 6  |-  ( ( dom  F  i^i  dom  G )  i^i  D ) 
C_  D
21sseli 3153 . . . . 5  |-  ( x  e.  ( ( dom 
F  i^i  dom  G )  i^i  D )  ->  x  e.  D )
3 fvres 5541 . . . . . 6  |-  ( x  e.  D  ->  (
( F  |`  D ) `
 x )  =  ( F `  x
) )
4 fvres 5541 . . . . . 6  |-  ( x  e.  D  ->  (
( G  |`  D ) `
 x )  =  ( G `  x
) )
53, 4oveq12d 5895 . . . . 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 4091 . . 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 3354 . . . . 5  |-  ( D  i^i  ( dom  F  i^i  dom  G ) )  =  ( ( D  i^i  dom  F )  i^i  ( D  i^i  dom  G ) )
9 incom 3329 . . . . 5  |-  ( ( dom  F  i^i  dom  G )  i^i  D )  =  ( D  i^i  ( dom  F  i^i  dom  G ) )
10 dmres 4930 . . . . . 6  |-  dom  ( F  |`  D )  =  ( D  i^i  dom  F )
11 dmres 4930 . . . . . 6  |-  dom  ( G  |`  D )  =  ( D  i^i  dom  G )
1210, 11ineq12i 3336 . . . . 5  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( D  i^i  dom 
F )  i^i  ( D  i^i  dom  G )
)
138, 9, 123eqtr4ri 2209 . . . 4  |-  ( dom  ( F  |`  D )  i^i  dom  ( G  |`  D ) )  =  ( ( dom  F  i^i  dom  G )  i^i 
D )
14 eqid 2177 . . . 4  |-  ( ( ( F  |`  D ) `
 x ) R ( ( G  |`  D ) `  x
) )  =  ( ( ( F  |`  D ) `  x
) R ( ( G  |`  D ) `  x ) )
1513, 14mpteq12i 4093 . . 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 4958 . . 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 2209 . 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 6137 . . 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 4908 . 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 4949 . . 3  |-  ( F  e.  V  ->  ( F  |`  D )  e. 
_V )
21 resexg 4949 . . 3  |-  ( G  e.  W  ->  ( G  |`  D )  e. 
_V )
22 offval3 6137 . . 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 289 . 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 2236 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 104    = wceq 1353    e. wcel 2148   _Vcvv 2739    i^i cin 3130    |-> cmpt 4066   dom cdm 4628    |` cres 4630   ` cfv 5218  (class class class)co 5877    oFcof 6083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-of 6085
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator