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Theorem cocnv 25542
Description: Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
cocnv  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )

Proof of Theorem cocnv
StepHypRef Expression
1 coass 5228 . 2  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
2 funcocnv2 5536 . . . . 5  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
32adantl 452 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
43coeq2d 4883 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
5 resco 5214 . . . 4  |-  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  o.  (  _I  |`  ran  G ) )
6 funrel 5309 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
7 coi1 5225 . . . . . . 7  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
86, 7syl 15 . . . . . 6  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
98reseq1d 4991 . . . . 5  |-  ( Fun 
F  ->  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  |`  ran  G
) )
109adantr 451 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  |`  ran  G
) )
115, 10syl5eqr 2362 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G ) )
124, 11eqtrd 2348 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  |`  ran  G
) )
131, 12syl5eq 2360 1  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    _I cid 4341   `'ccnv 4725   ran crn 4727    |` cres 4728    o. ccom 4730   Rel wrel 4731   Fun wfun 5286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-fun 5294
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