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Theorem ofres 6122
Description: Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
Hypotheses
Ref Expression
ofres.1  |-  ( ph  ->  F  Fn  A )
ofres.2  |-  ( ph  ->  G  Fn  B )
ofres.3  |-  ( ph  ->  A  e.  V )
ofres.4  |-  ( ph  ->  B  e.  W )
ofres.5  |-  ( A  i^i  B )  =  C
Assertion
Ref Expression
ofres  |-  ( ph  ->  ( F  oF R G )  =  ( ( F  |`  C )  oF R ( G  |`  C ) ) )

Proof of Theorem ofres
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ofres.1 . . 3  |-  ( ph  ->  F  Fn  A )
2 ofres.2 . . 3  |-  ( ph  ->  G  Fn  B )
3 ofres.3 . . 3  |-  ( ph  ->  A  e.  V )
4 ofres.4 . . 3  |-  ( ph  ->  B  e.  W )
5 ofres.5 . . 3  |-  ( A  i^i  B )  =  C
6 eqidd 2190 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2190 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 6115 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
9 inss1 3370 . . . . 5  |-  ( A  i^i  B )  C_  A
105, 9eqsstrri 3203 . . . 4  |-  C  C_  A
11 fnssres 5348 . . . 4  |-  ( ( F  Fn  A  /\  C  C_  A )  -> 
( F  |`  C )  Fn  C )
121, 10, 11sylancl 413 . . 3  |-  ( ph  ->  ( F  |`  C )  Fn  C )
13 inss2 3371 . . . . 5  |-  ( A  i^i  B )  C_  B
145, 13eqsstrri 3203 . . . 4  |-  C  C_  B
15 fnssres 5348 . . . 4  |-  ( ( G  Fn  B  /\  C  C_  B )  -> 
( G  |`  C )  Fn  C )
162, 14, 15sylancl 413 . . 3  |-  ( ph  ->  ( G  |`  C )  Fn  C )
17 ssexg 4157 . . . 4  |-  ( ( C  C_  A  /\  A  e.  V )  ->  C  e.  _V )
1810, 3, 17sylancr 414 . . 3  |-  ( ph  ->  C  e.  _V )
19 inidm 3359 . . 3  |-  ( C  i^i  C )  =  C
20 fvres 5558 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
2120adantl 277 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
22 fvres 5558 . . . 4  |-  ( x  e.  C  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2322adantl 277 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2412, 16, 18, 18, 19, 21, 23offval 6115 . 2  |-  ( ph  ->  ( ( F  |`  C )  oF R ( G  |`  C ) )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
258, 24eqtr4d 2225 1  |-  ( ph  ->  ( F  oF R G )  =  ( ( F  |`  C )  oF R ( G  |`  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752    i^i cin 3143    C_ wss 3144    |-> cmpt 4079    |` cres 4646    Fn wfn 5230   ` cfv 5235  (class class class)co 5897    oFcof 6105
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-of 6107
This theorem is referenced by: (None)
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