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Theorem ofresid 24008
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  o F R G )  =  ( F  o F ( 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 5829 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  B )
3 ofresid.2 . . . . . . . . 9  |-  ( ph  ->  G : A --> B )
43ffvelrnda 5829 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  e.  B )
52, 4jca 519 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  e.  B  /\  ( G `  x )  e.  B ) )
6 opelxp 4867 . . . . . . 7  |-  ( <.
( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  B
)  <->  ( ( F `
 x )  e.  B  /\  ( G `
 x )  e.  B ) )
75, 6sylibr 204 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  B ) )
8 fvres 5704 . . . . . 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 2409 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( R `  <. ( F `
 x ) ,  ( G `  x
) >. )  =  ( ( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. ) )
11 df-ov 6043 . . . 4  |-  ( ( F `  x ) R ( G `  x ) )  =  ( R `  <. ( F `  x ) ,  ( G `  x ) >. )
12 df-ov 6043 . . . 4  |-  ( ( F `  x ) ( R  |`  ( B  X.  B ) ) ( G `  x
) )  =  ( ( R  |`  ( B  X.  B ) ) `
 <. ( F `  x ) ,  ( G `  x )
>. )
1310, 11, 123eqtr4g 2461 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 x ) ( R  |`  ( B  X.  B ) ) ( G `  x ) ) )
1413mpteq2dva 4255 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  A  |->  ( ( F `  x
) ( R  |`  ( B  X.  B
) ) ( G `
 x ) ) ) )
15 ffn 5550 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
161, 15syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
17 ffn 5550 . . . 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 3510 . . 3  |-  ( A  i^i  A )  =  A
21 eqidd 2405 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
22 eqidd 2405 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
2316, 18, 19, 19, 20, 21, 22offval 6271 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) ) )
2416, 18, 19, 19, 20, 21, 22offval 6271 . 2  |-  ( ph  ->  ( F  o F ( R  |`  ( B  X.  B ) ) G )  =  ( x  e.  A  |->  ( ( F `  x
) ( R  |`  ( B  X.  B
) ) ( G `
 x ) ) ) )
2514, 23, 243eqtr4d 2446 1  |-  ( ph  ->  ( F  o F R G )  =  ( F  o F ( R  |`  ( B  X.  B ) ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3777    e. cmpt 4226    X. cxp 4835    |` cres 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262
This theorem is referenced by:  sitmcl  24616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264
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