Theorem fdmrn 6737

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 3959	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6540	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 722	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2763	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6539	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 722	. 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 1570   ⊆ wss 3905  dom cdm 5661  ran crn 5662  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-ss 3922  df-fn 6539  df-f 6540

This theorem is referenced by:  funcofd  6738  nvof1o  7278  usgrwwlks2on  30307  umgrwwlks2on  30308  rinvf1o  32975  smatrcl  34186  locfinref  34231  lfuhgr  35610  limccog  46336  funfocofob  47815  grimuhgr  48652  isuspgrim0lem  48658