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Theorem fdmrn 6102
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 3657 . . 3 ran 𝐹 ⊆ ran 𝐹
2 df-f 5930 . . 3 (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
31, 2mpbiran2 974 . 2 (𝐹:dom 𝐹⟶ran 𝐹𝐹 Fn dom 𝐹)
4 eqid 2651 . . 3 dom 𝐹 = dom 𝐹
5 df-fn 5929 . . 3 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 974 . 2 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
73, 6bitr2i 265 1 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wss 3607  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-in 3614  df-ss 3621  df-fn 5929  df-f 5930
This theorem is referenced by:  nvof1o  6576  umgrwwlks2on  26923  rinvf1o  29560  smatrcl  29990  locfinref  30036  fco3  39735  limccog  40170
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