MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcocnv2 Unicode version

Theorem funcocnv2 5498
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 5257 . . 3  |-  ( Fun 
F  <->  ( Rel  F  /\  ( F  o.  `' F )  C_  _I  ) )
21simprbi 450 . 2  |-  ( Fun 
F  ->  ( F  o.  `' F )  C_  _I  )
3 iss 4998 . . 3  |-  ( ( F  o.  `' F
)  C_  _I  <->  ( F  o.  `' F )  =  (  _I  |`  dom  ( F  o.  `' F ) ) )
4 dfdm4 4872 . . . . . . . 8  |-  dom  F  =  ran  `' F
5 dmcoeq 4947 . . . . . . . 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 4700 . . . . . . 7  |-  ran  F  =  dom  `' F
86, 7eqtr4i 2306 . . . . . 6  |-  dom  ( F  o.  `' F
)  =  ran  F
98a1i 10 . . . . 5  |-  ( Fun 
F  ->  dom  ( F  o.  `' F )  =  ran  F )
109reseq2d 4955 . . . 4  |-  ( Fun 
F  ->  (  _I  |` 
dom  ( F  o.  `' F ) )  =  (  _I  |`  ran  F
) )
1110eqeq2d 2294 . . 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 1623    C_ wss 3152    _I cid 4304   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  fococnv2  5499  f1cocnv2  5501  funcoeqres  5504  cnvcanOLD  26392  cocnv  26393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-fun 5257
  Copyright terms: Public domain W3C validator