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Theorem fdmrn 24039
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 3367 . . 3  |-  ran  F  C_ 
ran  F
2 df-f 5458 . . 3  |-  ( F : dom  F --> ran  F  <->  ( F  Fn  dom  F  /\  ran  F  C_  ran  F ) )
31, 2mpbiran2 886 . 2  |-  ( F : dom  F --> ran  F  <->  F  Fn  dom  F )
4 eqid 2436 . . 3  |-  dom  F  =  dom  F
5 df-fn 5457 . . 3  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
64, 5mpbiran2 886 . 2  |-  ( F  Fn  dom  F  <->  Fun  F )
73, 6bitr2i 242 1  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    C_ wss 3320   dom cdm 4878   ran crn 4879   Fun wfun 5448    Fn wfn 5449   -->wf 5450
This theorem is referenced by:  nvof1o  24040  rinvf1o  24042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-in 3327  df-ss 3334  df-fn 5457  df-f 5458
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