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

Theorem fdmrn 6768
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 4018 . . 3 ran 𝐹 ⊆ ran 𝐹
2 df-f 6567 . . 3 (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
31, 2mpbiran2 710 . 2 (𝐹:dom 𝐹⟶ran 𝐹𝐹 Fn dom 𝐹)
4 eqid 2735 . . 3 dom 𝐹 = dom 𝐹
5 df-fn 6566 . . 3 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 710 . 2 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
73, 6bitr2i 276 1 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wss 3963  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ss 3980  df-fn 6566  df-f 6567
This theorem is referenced by:  funcofd  6769  nvof1o  7300  umgrwwlks2on  29987  rinvf1o  32647  smatrcl  33757  locfinref  33802  lfuhgr  35102  limccog  45576  funfocofob  47028  isuspgrim0lem  47809  grimuhgr  47816
  Copyright terms: Public domain W3C validator