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Theorem fdmrn 5977
 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 3587 . . 3 ran 𝐹 ⊆ ran 𝐹
2 df-f 5808 . . 3 (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
31, 2mpbiran2 956 . 2 (𝐹:dom 𝐹⟶ran 𝐹𝐹 Fn dom 𝐹)
4 eqid 2610 . . 3 dom 𝐹 = dom 𝐹
5 df-fn 5807 . . 3 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 956 . 2 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
73, 6bitr2i 264 1 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ⊆ wss 3540  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-fn 5807  df-f 5808 This theorem is referenced by:  nvof1o  6436  rinvf1o  28814  smatrcl  29190  locfinref  29236  fco3  38416  limccog  38687  umgrwwlks2on  41161
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