Theorem fdmrn 6523

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 3914	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6339	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 709	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2758	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6338	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 709	. 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7	3, 6	bitr2i 279	1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

Syntax hints:   ↔ wb 209   = wceq 1538   ⊆ wss 3858  dom cdm 5524  ran crn 5525  Fun wfun 6329   Fn wfn 6330  ⟶wf 6331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-fn 6338  df-f 6339

This theorem is referenced by:  nvof1o  7029  umgrwwlks2on  27842  rinvf1o  30487  smatrcl  31267  locfinref  31312  lfuhgr  32595  fco3  42225  limccog  42628