MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fdmrn Structured version   Visualization version   GIF version

Theorem fdmrn 6736
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 4000 . . 3 ran 𝐹 ⊆ ran 𝐹
2 df-f 6536 . . 3 (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
31, 2mpbiran2 708 . 2 (𝐹:dom 𝐹⟶ran 𝐹𝐹 Fn dom 𝐹)
4 eqid 2731 . . 3 dom 𝐹 = dom 𝐹
5 df-fn 6535 . . 3 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 708 . 2 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
73, 6bitr2i 275 1 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wss 3944  dom cdm 5669  ran crn 5670  Fun wfun 6526   Fn wfn 6527  wf 6528
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3951  df-ss 3961  df-fn 6535  df-f 6536
This theorem is referenced by:  funcofd  6737  fco3OLD  6738  nvof1o  7262  umgrwwlks2on  29076  rinvf1o  31722  smatrcl  32607  locfinref  32652  lfuhgr  33939  limccog  44109  funfocofob  45558
  Copyright terms: Public domain W3C validator