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

Theorem funcoeqres 5284
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 5278 . . . 4  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
21coeq2d 4598 . . 3  |-  ( Fun 
G  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
3 coass 4949 . . . 4  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
43eqcomi 2092 . . 3  |-  ( F  o.  ( G  o.  `' G ) )  =  ( ( F  o.  G )  o.  `' G )
5 coires1 4948 . . 3  |-  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G )
62, 4, 53eqtr3g 2143 . 2  |-  ( Fun 
G  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
7 coeq1 4593 . 2  |-  ( ( F  o.  G )  =  H  ->  (
( F  o.  G
)  o.  `' G
)  =  ( H  o.  `' G ) )
86, 7sylan9req 2141 1  |-  ( ( Fun  G  /\  ( F  o.  G )  =  H )  ->  ( F  |`  ran  G )  =  ( H  o.  `' G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    _I cid 4115   `'ccnv 4437   ran crn 4439    |` cres 4440    o. ccom 4442   Fun wfun 5009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-fun 5017
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator