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Theorem ofres 6075
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 2171 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2171 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 6068 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
9 inss1 3347 . . . . 5  |-  ( A  i^i  B )  C_  A
105, 9eqsstrri 3180 . . . 4  |-  C  C_  A
11 fnssres 5311 . . . 4  |-  ( ( F  Fn  A  /\  C  C_  A )  -> 
( F  |`  C )  Fn  C )
121, 10, 11sylancl 411 . . 3  |-  ( ph  ->  ( F  |`  C )  Fn  C )
13 inss2 3348 . . . . 5  |-  ( A  i^i  B )  C_  B
145, 13eqsstrri 3180 . . . 4  |-  C  C_  B
15 fnssres 5311 . . . 4  |-  ( ( G  Fn  B  /\  C  C_  B )  -> 
( G  |`  C )  Fn  C )
162, 14, 15sylancl 411 . . 3  |-  ( ph  ->  ( G  |`  C )  Fn  C )
17 ssexg 4128 . . . 4  |-  ( ( C  C_  A  /\  A  e.  V )  ->  C  e.  _V )
1810, 3, 17sylancr 412 . . 3  |-  ( ph  ->  C  e.  _V )
19 inidm 3336 . . 3  |-  ( C  i^i  C )  =  C
20 fvres 5520 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
2120adantl 275 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
22 fvres 5520 . . . 4  |-  ( x  e.  C  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2322adantl 275 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( G  |`  C ) `
 x )  =  ( G `  x
) )
2412, 16, 18, 18, 19, 21, 23offval 6068 . 2  |-  ( ph  ->  ( ( F  |`  C )  oF R ( G  |`  C ) )  =  ( x  e.  C  |->  ( ( F `  x ) R ( G `  x ) ) ) )
258, 24eqtr4d 2206 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 1348    e. wcel 2141   _Vcvv 2730    i^i cin 3120    C_ wss 3121    |-> cmpt 4050    |` cres 4613    Fn wfn 5193   ` cfv 5198  (class class class)co 5853    oFcof 6059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061
This theorem is referenced by: (None)
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