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Theorem ofres 5753
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 2057 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2057 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 5747 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
9 inss1 3185 . . . . 5  |-  ( A  i^i  B )  C_  A
105, 9eqsstr3i 3004 . . . 4  |-  C  C_  A
11 fnssres 5040 . . . 4  |-  ( ( F  Fn  A  /\  C  C_  A )  -> 
( F  |`  C )  Fn  C )
121, 10, 11sylancl 398 . . 3  |-  ( ph  ->  ( F  |`  C )  Fn  C )
13 inss2 3186 . . . . 5  |-  ( A  i^i  B )  C_  B
145, 13eqsstr3i 3004 . . . 4  |-  C  C_  B
15 fnssres 5040 . . . 4  |-  ( ( G  Fn  B  /\  C  C_  B )  -> 
( G  |`  C )  Fn  C )
162, 14, 15sylancl 398 . . 3  |-  ( ph  ->  ( G  |`  C )  Fn  C )
17 ssexg 3924 . . . 4  |-  ( ( C  C_  A  /\  A  e.  V )  ->  C  e.  _V )
1810, 3, 17sylancr 399 . . 3  |-  ( ph  ->  C  e.  _V )
19 inidm 3174 . . 3  |-  ( C  i^i  C )  =  C
20 fvres 5226 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
2120adantl 266 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
22 fvres 5226 . . . 4  |-  ( x  e.  C  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2322adantl 266 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2412, 16, 18, 18, 19, 21, 23offval 5747 . 2  |-  ( ph  ->  ( ( F  |`  C )  oF R ( G  |`  C ) )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
258, 24eqtr4d 2091 1  |-  ( ph  ->  ( F  oF R G )  =  ( ( F  |`  C )  oF R ( G  |`  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574    i^i cin 2944    C_ wss 2945    |-> cmpt 3846    |` cres 4375    Fn wfn 4925   ` cfv 4930  (class class class)co 5540    oFcof 5738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-of 5740
This theorem is referenced by: (None)
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