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Theorem fdmrn 6523
 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 3914 . . 3 ran 𝐹 ⊆ ran 𝐹
2 df-f 6339 . . 3 (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
31, 2mpbiran2 709 . 2 (𝐹:dom 𝐹⟶ran 𝐹𝐹 Fn dom 𝐹)
4 eqid 2758 . . 3 dom 𝐹 = dom 𝐹
5 df-fn 6338 . . 3 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 709 . 2 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
73, 6bitr2i 279 1 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ⊆ wss 3858  dom cdm 5524  ran crn 5525  Fun wfun 6329   Fn wfn 6330  ⟶wf 6331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-fn 6338  df-f 6339 This theorem is referenced by:  nvof1o  7029  umgrwwlks2on  27842  rinvf1o  30487  smatrcl  31267  locfinref  31312  lfuhgr  32595  fco3  42225  limccog  42628
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