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Theorem funcocnv2 5729
Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
funcocnv2  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )

Proof of Theorem funcocnv2
StepHypRef Expression
1 df-fun 5485 . . 3  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
21simprbi 452 . 2  |-  ( Fun 
F  ->  ( F  o.  `' F )  C_  _I  )
3 iss 5218 . . 3  |-  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  dom  ( F  o.  `' F ) ) )
4 dfdm4 5092 . . . . . . . 8  |-  dom  F  =  ran  `' F
5 dmcoeq 5167 . . . . . . . 8  |-  ( dom 
F  =  ran  `' F  ->  dom  ( F  o.  `' F )  =  dom  `' F )
64, 5ax-mp 5 . . . . . . 7  |-  dom  ( F  o.  `' F
)  =  dom  `' F
7 df-rn 4918 . . . . . . 7  |-  ran  F  =  dom  `' F
86, 7eqtr4i 2465 . . . . . 6  |-  dom  ( F  o.  `' F
)  =  ran  F
98a1i 11 . . . . 5  |-  ( Fun 
F  ->  dom  ( F  o.  `' F )  =  ran  F )
109reseq2d 5175 . . . 4  |-  ( Fun 
F  ->  (  _I  |` 
dom  ( F  o.  `' F ) )  =  (  _I  |`  ran  F
) )
1110eqeq2d 2453 . . 3  |-  ( Fun 
F  ->  ( ( F  o.  `' F
)  =  (  _I  |`  dom  ( F  o.  `' F ) )  <->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) ) )
123, 11syl5bb 250 . 2  |-  ( Fun 
F  ->  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) ) )
132, 12mpbid 203 1  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    C_ wss 3306    _I cid 4522   `'ccnv 4906   dom cdm 4907   ran crn 4908    |` cres 4909    o. ccom 4911   Rel wrel 4912   Fun wfun 5477
This theorem is referenced by:  fococnv2  5730  f1cocnv2  5732  funcoeqres  5735  cocnv  26465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-fun 5485
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