ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funcoeqres Unicode version

Theorem funcoeqres 5366
Description: 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 5360 . . . 4  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
21coeq2d 4671 . . 3  |-  ( Fun 
G  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
3 coass 5027 . . . 4  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
43eqcomi 2121 . . 3  |-  ( F  o.  ( G  o.  `' G ) )  =  ( ( F  o.  G )  o.  `' G )
5 coires1 5026 . . 3  |-  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G )
62, 4, 53eqtr3g 2173 . 2  |-  ( Fun 
G  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
7 coeq1 4666 . 2  |-  ( ( F  o.  G )  =  H  ->  (
( F  o.  G
)  o.  `' G
)  =  ( H  o.  `' G ) )
86, 7sylan9req 2171 1  |-  ( ( Fun  G  /\  ( F  o.  G )  =  H )  ->  ( F  |`  ran  G )  =  ( H  o.  `' G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    _I cid 4180   `'ccnv 4508   ran crn 4510    |` cres 4511    o. ccom 4513   Fun wfun 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-fun 5095
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator