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Theorem cocnv 26427
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 5388 . 2  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
2 funcocnv2 5700 . . . . 5  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
32adantl 453 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
43coeq2d 5035 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
5 resco 5374 . . . 4  |-  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  o.  (  _I  |`  ran  G ) )
6 funrel 5471 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
7 coi1 5385 . . . . . . 7  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
86, 7syl 16 . . . . . 6  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
98reseq1d 5145 . . . . 5  |-  ( Fun 
F  ->  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  |`  ran  G
) )
109adantr 452 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  |`  ran  G
) )
115, 10syl5eqr 2482 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G ) )
124, 11eqtrd 2468 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  |`  ran  G
) )
131, 12syl5eq 2480 1  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    _I cid 4493   `'ccnv 4877   ran crn 4879    |` cres 4880    o. ccom 4882   Rel wrel 4883   Fun wfun 5448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-fun 5456
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