Theorem fdmrn 6741

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 6544	. . 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 2765	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6543	. . 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 3906  dom cdm 5663  ran crn 5664  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536

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 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-ss 3923  df-fn 6543  df-f 6544

This theorem is used by:  funcofd  6742  nvof1o  7284  lfuhgr  29529  usgrwwlks2on  30350  umgrwwlks2on  30351  rinvf1o  33022  smatrcl  34226  locfinref  34271  limccog  46369  funfocofob  47848  grimuhgr  48685  isuspgrim0lem  48691