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Theorem ofres 6110
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  o F R G )  =  ( ( F  |`  C )  o F 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 2297 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2297 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 6101 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
9 inss1 3402 . . . . 5  |-  ( A  i^i  B )  C_  A
105, 9eqsstr3i 3222 . . . 4  |-  C  C_  A
11 fnssres 5373 . . . 4  |-  ( ( F  Fn  A  /\  C  C_  A )  -> 
( F  |`  C )  Fn  C )
121, 10, 11sylancl 643 . . 3  |-  ( ph  ->  ( F  |`  C )  Fn  C )
13 inss2 3403 . . . . 5  |-  ( A  i^i  B )  C_  B
145, 13eqsstr3i 3222 . . . 4  |-  C  C_  B
15 fnssres 5373 . . . 4  |-  ( ( G  Fn  B  /\  C  C_  B )  -> 
( G  |`  C )  Fn  C )
162, 14, 15sylancl 643 . . 3  |-  ( ph  ->  ( G  |`  C )  Fn  C )
17 ssexg 4176 . . . 4  |-  ( ( C  C_  A  /\  A  e.  V )  ->  C  e.  _V )
1810, 3, 17sylancr 644 . . 3  |-  ( ph  ->  C  e.  _V )
19 inidm 3391 . . 3  |-  ( C  i^i  C )  =  C
20 fvres 5558 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
2120adantl 452 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
22 fvres 5558 . . . 4  |-  ( x  e.  C  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2322adantl 452 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2412, 16, 18, 18, 19, 21, 23offval 6101 . 2  |-  ( ph  ->  ( ( F  |`  C )  o F R ( G  |`  C ) )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
258, 24eqtr4d 2331 1  |-  ( ph  ->  ( F  o F R G )  =  ( ( F  |`  C )  o F R ( G  |`  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165    e. cmpt 4093    |` cres 4707    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094
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