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Theorem cocnv 30456
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 5509 . 2  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
2 funcocnv2 5822 . . . . 5  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
32adantl 464 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
43coeq2d 5154 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
5 resco 5494 . . . 4  |-  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  o.  (  _I  |`  ran  G ) )
6 funrel 5587 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
7 coi1 5506 . . . . . . 7  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
86, 7syl 16 . . . . . 6  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
98reseq1d 5261 . . . . 5  |-  ( Fun 
F  ->  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  |`  ran  G
) )
109adantr 463 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  |`  ran  G
) )
115, 10syl5eqr 2509 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G ) )
124, 11eqtrd 2495 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  |`  ran  G
) )
131, 12syl5eq 2507 1  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    _I cid 4779   `'ccnv 4987   ran crn 4989    |` cres 4990    o. ccom 4992   Rel wrel 4993   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-fun 5572
This theorem is referenced by: (None)
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