Theorem fdmrn 6693

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 3956	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6496	. . 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 2736	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6495	. . 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 3901  dom cdm 5624  ran crn 5625  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2728  df-ss 3918  df-fn 6495  df-f 6496

This theorem is referenced by:  funcofd  6694  nvof1o  7226  usgrwwlks2on  30031  umgrwwlks2on  30032  rinvf1o  32708  smatrcl  33953  locfinref  33998  lfuhgr  35312  limccog  45866  funfocofob  47324  grimuhgr  48133  isuspgrim0lem  48139