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Theorem ofres 5907
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 2096 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2096 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 5901 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
9 inss1 3235 . . . . 5  |-  ( A  i^i  B )  C_  A
105, 9eqsstr3i 3072 . . . 4  |-  C  C_  A
11 fnssres 5161 . . . 4  |-  ( ( F  Fn  A  /\  C  C_  A )  -> 
( F  |`  C )  Fn  C )
121, 10, 11sylancl 405 . . 3  |-  ( ph  ->  ( F  |`  C )  Fn  C )
13 inss2 3236 . . . . 5  |-  ( A  i^i  B )  C_  B
145, 13eqsstr3i 3072 . . . 4  |-  C  C_  B
15 fnssres 5161 . . . 4  |-  ( ( G  Fn  B  /\  C  C_  B )  -> 
( G  |`  C )  Fn  C )
162, 14, 15sylancl 405 . . 3  |-  ( ph  ->  ( G  |`  C )  Fn  C )
17 ssexg 3999 . . . 4  |-  ( ( C  C_  A  /\  A  e.  V )  ->  C  e.  _V )
1810, 3, 17sylancr 406 . . 3  |-  ( ph  ->  C  e.  _V )
19 inidm 3224 . . 3  |-  ( C  i^i  C )  =  C
20 fvres 5364 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
2120adantl 272 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
22 fvres 5364 . . . 4  |-  ( x  e.  C  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2322adantl 272 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2412, 16, 18, 18, 19, 21, 23offval 5901 . 2  |-  ( ph  ->  ( ( F  |`  C )  oF R ( G  |`  C ) )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
258, 24eqtr4d 2130 1  |-  ( ph  ->  ( F  oF R G )  =  ( ( F  |`  C )  oF R ( G  |`  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1296    e. wcel 1445   _Vcvv 2633    i^i cin 3012    C_ wss 3013    |-> cmpt 3921    |` cres 4469    Fn wfn 5044   ` cfv 5049  (class class class)co 5690    oFcof 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-of 5894
This theorem is referenced by: (None)
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