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Theorem funcoeqres 5780
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 5774 . . . 4  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
21coeq2d 5111 . . 3  |-  ( Fun 
G  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
3 coass 5465 . . . 4  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
43eqcomi 2467 . . 3  |-  ( F  o.  ( G  o.  `' G ) )  =  ( ( F  o.  G )  o.  `' G )
5 coires1 5464 . . 3  |-  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G )
62, 4, 53eqtr3g 2518 . 2  |-  ( Fun 
G  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
7 coeq1 5106 . 2  |-  ( ( F  o.  G )  =  H  ->  (
( F  o.  G
)  o.  `' G
)  =  ( H  o.  `' G ) )
86, 7sylan9req 2516 1  |-  ( ( Fun  G  /\  ( F  o.  G )  =  H )  ->  ( F  |`  ran  G )  =  ( H  o.  `' G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    _I cid 4740   `'ccnv 4948   ran crn 4950    |` cres 4951    o. ccom 4953   Fun wfun 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-fun 5529
This theorem is referenced by:  evlseu  17727  frlmup4  18355
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