Theorem fdmrn 6771

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 3935	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6570	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 723	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2763	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6569	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 723	. 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7	3, 6	bitr2i 279	1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ⊆ wss 3900  dom cdm 5658  ran crn 5659  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ss 3901  df-fn 6569  df-f 6570

This theorem is used by:  funcofd  6772  nvof1o  7319  lfuhgr  29783  usgrwwlks2on  30604  umgrwwlks2on  30605  rinvf1o  33322  smatrcl  34539  locfinref  34584  limccog  46905  funfocofob  48421  grimuhgr  49258  isuspgrim0lem  49264