Theorem fdmrn 6686

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 3937	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6489	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 716	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2739	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6488	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 716	. 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7	3, 6	bitr2i 277	1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

Syntax hints:   ↔ wb 207   = wceq 1547   ⊆ wss 3883  dom cdm 5618  ran crn 5619  Fun wfun 6479   Fn wfn 6480  ⟶wf 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-ss 3900  df-fn 6488  df-f 6489

This theorem is referenced by:  funcofd  6687  nvof1o  7224  usgrwwlks2on  30044  umgrwwlks2on  30045  rinvf1o  32722  smatrcl  33980  locfinref  34025  lfuhgr  35346  limccog  46065  funfocofob  47541  grimuhgr  48378  isuspgrim0lem  48384