Theorem fdmrn 6737

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 3981	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6535	. . 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 2735	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6534	. . 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 3926  dom cdm 5654  ran crn 5655  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527

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 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-ss 3943  df-fn 6534  df-f 6535

This theorem is referenced by:  funcofd  6738  nvof1o  7273  umgrwwlks2on  29939  rinvf1o  32608  smatrcl  33827  locfinref  33872  lfuhgr  35140  limccog  45649  funfocofob  47107  grimuhgr  47900  isuspgrim0lem  47906