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Theorem funcoeqres 5754
Description: Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
funcoeqres  |-  ( ( Fun  G  /\  ( F  o.  G )  =  H )  ->  ( F  |`  ran  G )  =  ( H  o.  `' G ) )

Proof of Theorem funcoeqres
StepHypRef Expression
1 funcocnv2 5748 . . . 4  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
21coeq2d 5078 . . 3  |-  ( Fun 
G  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
3 coass 5434 . . . 4  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
43eqcomi 2395 . . 3  |-  ( F  o.  ( G  o.  `' G ) )  =  ( ( F  o.  G )  o.  `' G )
5 coires1 5433 . . 3  |-  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G )
62, 4, 53eqtr3g 2446 . 2  |-  ( Fun 
G  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
7 coeq1 5073 . 2  |-  ( ( F  o.  G )  =  H  ->  (
( F  o.  G
)  o.  `' G
)  =  ( H  o.  `' G ) )
86, 7sylan9req 2444 1  |-  ( ( Fun  G  /\  ( F  o.  G )  =  H )  ->  ( F  |`  ran  G )  =  ( H  o.  `' G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    _I cid 4704   `'ccnv 4912   ran crn 4914    |` cres 4915    o. ccom 4917   Fun wfun 5490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-fun 5498
This theorem is referenced by:  evlseu  18298  frlmup4  18921
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