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Theorem fdmrn 23051
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 3210 . . 3  |-  ran  F  C_ 
ran  F
2 df-f 5275 . . 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 2296 . . 3  |-  dom  F  =  dom  F
5 df-fn 5274 . . 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 1632    C_ wss 3165   dom cdm 4705   ran crn 4706   Fun wfun 5265    Fn wfn 5266   -->wf 5267
This theorem is referenced by:  nvof1o  23052  rinvf1o  23054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-fn 5274  df-f 5275
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