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Theorem ofresid 25979
Description: Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
Hypotheses
Ref Expression
ofresid.1  |-  ( ph  ->  F : A --> B )
ofresid.2  |-  ( ph  ->  G : A --> B )
ofresid.3  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
ofresid  |-  ( ph  ->  ( F  oF R G )  =  ( F  oF ( R  |`  ( B  X.  B ) ) G ) )

Proof of Theorem ofresid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ofresid.1 . . . . . . . . 9  |-  ( ph  ->  F : A --> B )
21ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  B )
3 ofresid.2 . . . . . . . . 9  |-  ( ph  ->  G : A --> B )
43ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  e.  B )
52, 4jca 532 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  e.  B  /\  ( G `  x )  e.  B ) )
6 opelxp 4888 . . . . . . 7  |-  ( <.
( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  B
)  <->  ( ( F `
 x )  e.  B  /\  ( G `
 x )  e.  B ) )
75, 6sylibr 212 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  B ) )
8 fvres 5723 . . . . . 6  |-  ( <.
( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  B
)  ->  ( ( R  |`  ( B  X.  B ) ) `  <. ( F `  x
) ,  ( G `
 x ) >.
)  =  ( R `
 <. ( F `  x ) ,  ( G `  x )
>. ) )
97, 8syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. )  =  ( R `  <. ( F `
 x ) ,  ( G `  x
) >. ) )
109eqcomd 2448 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( R `  <. ( F `
 x ) ,  ( G `  x
) >. )  =  ( ( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. ) )
11 df-ov 6113 . . . 4  |-  ( ( F `  x ) R ( G `  x ) )  =  ( R `  <. ( F `  x ) ,  ( G `  x ) >. )
12 df-ov 6113 . . . 4  |-  ( ( F `  x ) ( R  |`  ( B  X.  B ) ) ( G `  x
) )  =  ( ( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. )
1310, 11, 123eqtr4g 2500 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 x ) ( R  |`  ( B  X.  B ) ) ( G `  x ) ) )
1413mpteq2dva 4397 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  A  |->  ( ( F `  x
) ( R  |`  ( B  X.  B
) ) ( G `
 x ) ) ) )
15 ffn 5578 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
161, 15syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
17 ffn 5578 . . . 4  |-  ( G : A --> B  ->  G  Fn  A )
183, 17syl 16 . . 3  |-  ( ph  ->  G  Fn  A )
19 ofresid.3 . . 3  |-  ( ph  ->  A  e.  V )
20 inidm 3578 . . 3  |-  ( A  i^i  A )  =  A
21 eqidd 2444 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
22 eqidd 2444 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
2316, 18, 19, 19, 20, 21, 22offval 6346 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) ) )
2416, 18, 19, 19, 20, 21, 22offval 6346 . 2  |-  ( ph  ->  ( F  oF ( R  |`  ( B  X.  B ) ) G )  =  ( x  e.  A  |->  ( ( F `  x
) ( R  |`  ( B  X.  B
) ) ( G `
 x ) ) ) )
2514, 23, 243eqtr4d 2485 1  |-  ( ph  ->  ( F  oF R G )  =  ( F  oF ( R  |`  ( B  X.  B ) ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3902    e. cmpt 4369    X. cxp 4857    |` cres 4861    Fn wfn 5432   -->wf 5433   ` cfv 5437  (class class class)co 6110    oFcof 6337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339
This theorem is referenced by: (None)
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