Theorem fdmrn 6742

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 3930	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6543	. . 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 2760	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6542	. . 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 1569   ⊆ wss 3895  dom cdm 5650  ran crn 5651  Fun wfun 6533   Fn wfn 6534  ⟶wf 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-ss 3896  df-fn 6542  df-f 6543

This theorem is used by:  funcofd  6743  nvof1o  7289  lfuhgr  29732  usgrwwlks2on  30553  umgrwwlks2on  30554  rinvf1o  33272  smatrcl  34475  locfinref  34520  limccog  46726  funfocofob  48242  grimuhgr  49079  isuspgrim0lem  49085