Theorem fdmrn 6687

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 3960	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6490	. . 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 2729	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6489	. . 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 1540   ⊆ wss 3905  dom cdm 5623  ran crn 5624  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-ss 3922  df-fn 6489  df-f 6490

This theorem is referenced by:  funcofd  6688  nvof1o  7221  umgrwwlks2on  29921  rinvf1o  32592  smatrcl  33782  locfinref  33827  lfuhgr  35110  limccog  45621  funfocofob  47082  grimuhgr  47891  isuspgrim0lem  47897