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Theorem funcocnv2 5151
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 4904 . . 3  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
21simprbi 260 . 2  |-  ( Fun 
F  ->  ( F  o.  `' F )  C_  _I  )
3 iss 4654 . . 3  |-  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  dom  ( F  o.  `' F ) ) )
4 dfdm4 4527 . . . . . . . 8  |-  dom  F  =  ran  `' F
5 dmcoeq 4604 . . . . . . . 8  |-  ( dom 
F  =  ran  `' F  ->  dom  ( F  o.  `' F )  =  dom  `' F )
64, 5ax-mp 7 . . . . . . 7  |-  dom  ( F  o.  `' F
)  =  dom  `' F
7 df-rn 4356 . . . . . . 7  |-  ran  F  =  dom  `' F
86, 7eqtr4i 2063 . . . . . 6  |-  dom  ( F  o.  `' F
)  =  ran  F
98a1i 9 . . . . 5  |-  ( Fun 
F  ->  dom  ( F  o.  `' F )  =  ran  F )
109reseq2d 4612 . . . 4  |-  ( Fun 
F  ->  (  _I  |` 
dom  ( F  o.  `' F ) )  =  (  _I  |`  ran  F
) )
1110eqeq2d 2051 . . 3  |-  ( Fun 
F  ->  ( ( F  o.  `' F
)  =  (  _I  |`  dom  ( F  o.  `' F ) )  <->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) ) )
123, 11syl5bb 181 . 2  |-  ( Fun 
F  ->  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) ) )
132, 12mpbid 135 1  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    C_ wss 2917    _I cid 4025   `'ccnv 4344   dom cdm 4345   ran crn 4346    |` cres 4347    o. ccom 4349   Rel wrel 4350   Fun wfun 4896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-fun 4904
This theorem is referenced by:  fococnv2  5152  f1cocnv2  5154  funcoeqres  5157
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