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Theorem fdmrn 23035
Description: A different way to write  F is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Assertion
Ref Expression
fdmrn  |-  ( Fun 
F  <->  F : dom  F --> ran  F )

Proof of Theorem fdmrn
StepHypRef Expression
1 ssid 3197 . . 3  |-  ran  F  C_ 
ran  F
2 df-f 5259 . . 3  |-  ( F : dom  F --> ran  F  <->  ( F  Fn  dom  F  /\  ran  F  C_  ran  F ) )
31, 2mpbiran2 885 . 2  |-  ( F : dom  F --> ran  F  <->  F  Fn  dom  F )
4 eqid 2283 . . 3  |-  dom  F  =  dom  F
5 df-fn 5258 . . 3  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
64, 5mpbiran2 885 . 2  |-  ( F  Fn  dom  F  <->  Fun  F )
73, 6bitr2i 241 1  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    C_ wss 3152   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  nvof1o  23036  rinvf1o  23038
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-fn 5258  df-f 5259
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