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Theorem cocnv 26393
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 5191 . 2  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
2 funcocnv2 5498 . . . . 5  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
32adantl 452 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
43coeq2d 4846 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
5 resco 5177 . . . 4  |-  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  o.  (  _I  |`  ran  G ) )
6 funrel 5272 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
7 coi1 5188 . . . . . . 7  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
86, 7syl 15 . . . . . 6  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
98reseq1d 4954 . . . . 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 2329 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G ) )
124, 11eqtrd 2315 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  |`  ran  G
) )
131, 12syl5eq 2327 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 1623    _I cid 4304   `'ccnv 4688   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694   Fun wfun 5249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-fun 5257
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