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Theorem ofresid 27938
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 . . . . . . . 8  |-  ( ph  ->  F : A --> B )
21ffvelrnda 6011 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  B )
3 ofresid.2 . . . . . . . 8  |-  ( ph  ->  G : A --> B )
43ffvelrnda 6011 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  e.  B )
5 opelxp 4855 . . . . . . 7  |-  ( <.
( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  B
)  <->  ( ( F `
 x )  e.  B  /\  ( G `
 x )  e.  B ) )
62, 4, 5sylanbrc 664 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  B ) )
7 fvres 5865 . . . . . 6  |-  ( <.
( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  B
)  ->  ( ( R  |`  ( B  X.  B ) ) `  <. ( F `  x
) ,  ( G `
 x ) >.
)  =  ( R `
 <. ( F `  x ) ,  ( G `  x )
>. ) )
86, 7syl 17 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. )  =  ( R `  <. ( F `
 x ) ,  ( G `  x
) >. ) )
98eqcomd 2412 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( R `  <. ( F `
 x ) ,  ( G `  x
) >. )  =  ( ( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. ) )
10 df-ov 6283 . . . 4  |-  ( ( F `  x ) R ( G `  x ) )  =  ( R `  <. ( F `  x ) ,  ( G `  x ) >. )
11 df-ov 6283 . . . 4  |-  ( ( F `  x ) ( R  |`  ( B  X.  B ) ) ( G `  x
) )  =  ( ( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. )
129, 10, 113eqtr4g 2470 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 x ) ( R  |`  ( B  X.  B ) ) ( G `  x ) ) )
1312mpteq2dva 4483 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  A  |->  ( ( F `  x
) ( R  |`  ( B  X.  B
) ) ( G `
 x ) ) ) )
14 ffn 5716 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
151, 14syl 17 . . 3  |-  ( ph  ->  F  Fn  A )
16 ffn 5716 . . . 4  |-  ( G : A --> B  ->  G  Fn  A )
173, 16syl 17 . . 3  |-  ( ph  ->  G  Fn  A )
18 ofresid.3 . . 3  |-  ( ph  ->  A  e.  V )
19 inidm 3650 . . 3  |-  ( A  i^i  A )  =  A
20 eqidd 2405 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
21 eqidd 2405 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
2215, 17, 18, 18, 19, 20, 21offval 6530 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) ) )
2315, 17, 18, 18, 19, 20, 21offval 6530 . 2  |-  ( ph  ->  ( F  oF ( R  |`  ( B  X.  B ) ) G )  =  ( x  e.  A  |->  ( ( F `  x
) ( R  |`  ( B  X.  B
) ) ( G `
 x ) ) ) )
2413, 22, 233eqtr4d 2455 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 1407    e. wcel 1844   <.cop 3980    |-> cmpt 4455    X. cxp 4823    |` cres 4827    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280    oFcof 6521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523
This theorem is referenced by: (None)
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