Theorem fdmrn 6690

Description: A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Assertion

Ref	Expression
fdmrn	⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

Proof of Theorem fdmrn

Step	Hyp	Ref	Expression
1		ssid 3953	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6493	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 710	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2733	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6492	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 710	. 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7	3, 6	bitr2i 276	1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

Syntax hints:   ↔ wb 206   = wceq 1541   ⊆ wss 3898  dom cdm 5621  ran crn 5622  Fun wfun 6483   Fn wfn 6484  ⟶wf 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2725  df-ss 3915  df-fn 6492  df-f 6493

This theorem is referenced by:  funcofd  6691  nvof1o  7223  usgrwwlks2on  29957  umgrwwlks2on  29958  rinvf1o  32634  smatrcl  33881  locfinref  33926  lfuhgr  35234  limccog  45782  funfocofob  47240  grimuhgr  48049  isuspgrim0lem  48055