Theorem fdmrn 6718

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 3956	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6520	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 720	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2761	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6519	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 720	. 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7	3, 6	bitr2i 278	1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

Syntax hints:   ↔ wb 208   = wceq 1559   ⊆ wss 3902  dom cdm 5643  ran crn 5644  Fun wfun 6510   Fn wfn 6511  ⟶wf 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-cleq 2753  df-ss 3919  df-fn 6519  df-f 6520

This theorem is referenced by:  funcofd  6719  nvof1o  7259  usgrwwlks2on  30115  umgrwwlks2on  30116  rinvf1o  32793  smatrcl  34054  locfinref  34099  lfuhgr  35429  limccog  46157  funfocofob  47633  grimuhgr  48470  isuspgrim0lem  48476