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Theorem funcoeqres 5698
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 5692 . . . 4  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
21coeq2d 5027 . . 3  |-  ( Fun 
G  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
3 coass 5380 . . . 4  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
43eqcomi 2439 . . 3  |-  ( F  o.  ( G  o.  `' G ) )  =  ( ( F  o.  G )  o.  `' G )
5 coires1 5379 . . 3  |-  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G )
62, 4, 53eqtr3g 2490 . 2  |-  ( Fun 
G  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
7 coeq1 5022 . 2  |-  ( ( F  o.  G )  =  H  ->  (
( F  o.  G
)  o.  `' G
)  =  ( H  o.  `' G ) )
86, 7sylan9req 2488 1  |-  ( ( Fun  G  /\  ( F  o.  G )  =  H )  ->  ( F  |`  ran  G )  =  ( H  o.  `' G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    _I cid 4485   `'ccnv 4869   ran crn 4871    |` cres 4872    o. ccom 4874   Fun wfun 5440
This theorem is referenced by:  evlseu  19929  frlmup4  27221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-fun 5448
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