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Theorem funcocnv2 5578
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 5336 . . 3  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
21simprbi 450 . 2  |-  ( Fun 
F  ->  ( F  o.  `' F )  C_  _I  )
3 iss 5077 . . 3  |-  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  dom  ( F  o.  `' F ) ) )
4 dfdm4 4951 . . . . . . . 8  |-  dom  F  =  ran  `' F
5 dmcoeq 5026 . . . . . . . 8  |-  ( dom 
F  =  ran  `' F  ->  dom  ( F  o.  `' F )  =  dom  `' F )
64, 5ax-mp 8 . . . . . . 7  |-  dom  ( F  o.  `' F
)  =  dom  `' F
7 df-rn 4779 . . . . . . 7  |-  ran  F  =  dom  `' F
86, 7eqtr4i 2381 . . . . . 6  |-  dom  ( F  o.  `' F
)  =  ran  F
98a1i 10 . . . . 5  |-  ( Fun 
F  ->  dom  ( F  o.  `' F )  =  ran  F )
109reseq2d 5034 . . . 4  |-  ( Fun 
F  ->  (  _I  |` 
dom  ( F  o.  `' F ) )  =  (  _I  |`  ran  F
) )
1110eqeq2d 2369 . . 3  |-  ( Fun 
F  ->  ( ( F  o.  `' F
)  =  (  _I  |`  dom  ( F  o.  `' F ) )  <->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) ) )
123, 11syl5bb 248 . 2  |-  ( Fun 
F  ->  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) ) )
132, 12mpbid 201 1  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    C_ wss 3228    _I cid 4383   `'ccnv 4767   dom cdm 4768   ran crn 4769    |` cres 4770    o. ccom 4772   Rel wrel 4773   Fun wfun 5328
This theorem is referenced by:  fococnv2  5579  f1cocnv2  5581  funcoeqres  5584  cnvcanOLD  25716  cocnv  25717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-fun 5336
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