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Theorem fdmrn 5963
Description: A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Assertion
Ref Expression
fdmrn (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)

Proof of Theorem fdmrn
StepHypRef Expression
1 ssid 3586 . . 3 ran 𝐹 ⊆ ran 𝐹
2 df-f 5794 . . 3 (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
31, 2mpbiran2 955 . 2 (𝐹:dom 𝐹⟶ran 𝐹𝐹 Fn dom 𝐹)
4 eqid 2609 . . 3 dom 𝐹 = dom 𝐹
5 df-fn 5793 . . 3 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 955 . 2 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
73, 6bitr2i 263 1 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wss 3539  dom cdm 5028  ran crn 5029  Fun wfun 5784   Fn wfn 5785  wf 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-in 3546  df-ss 3553  df-fn 5793  df-f 5794
This theorem is referenced by:  nvof1o  6414  rinvf1o  28620  smatrcl  28996  locfinref  29042  fco3  38219  limccog  38491  umgrwwlks2on  41163
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