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Theorem funcocnv2 5278
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 5017 . . 3  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
21simprbi 269 . 2  |-  ( Fun 
F  ->  ( F  o.  `' F )  C_  _I  )
3 iss 4758 . . 3  |-  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  dom  ( F  o.  `' F ) ) )
4 dfdm4 4628 . . . . . . . 8  |-  dom  F  =  ran  `' F
5 dmcoeq 4705 . . . . . . . 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 4449 . . . . . . 7  |-  ran  F  =  dom  `' F
86, 7eqtr4i 2111 . . . . . 6  |-  dom  ( F  o.  `' F
)  =  ran  F
98a1i 9 . . . . 5  |-  ( Fun 
F  ->  dom  ( F  o.  `' F )  =  ran  F )
109reseq2d 4713 . . . 4  |-  ( Fun 
F  ->  (  _I  |` 
dom  ( F  o.  `' F ) )  =  (  _I  |`  ran  F
) )
1110eqeq2d 2099 . . 3  |-  ( Fun 
F  ->  ( ( F  o.  `' F
)  =  (  _I  |`  dom  ( F  o.  `' F ) )  <->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) ) )
123, 11syl5bb 190 . 2  |-  ( Fun 
F  ->  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) ) )
132, 12mpbid 145 1  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2999    _I cid 4115   `'ccnv 4437   dom cdm 4438   ran crn 4439    |` cres 4440    o. ccom 4442   Rel wrel 4443   Fun wfun 5009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-fun 5017
This theorem is referenced by:  fococnv2  5279  f1cocnv2  5281  funcoeqres  5284
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