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Theorem ofresid 25895
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 5840 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  B )
3 ofresid.2 . . . . . . . . 9  |-  ( ph  ->  G : A --> B )
43ffvelrnda 5840 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  e.  B )
52, 4jca 529 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  e.  B  /\  ( G `  x )  e.  B ) )
6 opelxp 4865 . . . . . . 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 5701 . . . . . 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 2446 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( R `  <. ( F `
 x ) ,  ( G `  x
) >. )  =  ( ( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. ) )
11 df-ov 6093 . . . 4  |-  ( ( F `  x ) R ( G `  x ) )  =  ( R `  <. ( F `  x ) ,  ( G `  x ) >. )
12 df-ov 6093 . . . 4  |-  ( ( F `  x ) ( R  |`  ( B  X.  B ) ) ( G `  x
) )  =  ( ( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. )
1310, 11, 123eqtr4g 2498 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 x ) ( R  |`  ( B  X.  B ) ) ( G `  x ) ) )
1413mpteq2dva 4375 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  A  |->  ( ( F `  x
) ( R  |`  ( B  X.  B
) ) ( G `
 x ) ) ) )
15 ffn 5556 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
161, 15syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
17 ffn 5556 . . . 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 3556 . . 3  |-  ( A  i^i  A )  =  A
21 eqidd 2442 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
22 eqidd 2442 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
2316, 18, 19, 19, 20, 21, 22offval 6326 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) ) )
2416, 18, 19, 19, 20, 21, 22offval 6326 . 2  |-  ( ph  ->  ( F  oF ( R  |`  ( B  X.  B ) ) G )  =  ( x  e.  A  |->  ( ( F `  x
) ( R  |`  ( B  X.  B
) ) ( G `
 x ) ) ) )
2514, 23, 243eqtr4d 2483 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 1364    e. wcel 1761   <.cop 3880    e. cmpt 4347    X. cxp 4834    |` cres 4838    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319
This theorem is referenced by: (None)
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