Theorem fdmrn 6699

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 3944	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6502	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 711	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2736	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6501	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 711	. 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7	3, 6	bitr2i 276	1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

Syntax hints:   ↔ wb 206   = wceq 1542   ⊆ wss 3889  dom cdm 5631  ran crn 5632  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494

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 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2728  df-ss 3906  df-fn 6501  df-f 6502

This theorem is referenced by:  funcofd  6700  nvof1o  7235  usgrwwlks2on  30026  umgrwwlks2on  30027  rinvf1o  32703  smatrcl  33940  locfinref  33985  lfuhgr  35300  limccog  46050  funfocofob  47526  grimuhgr  48363  isuspgrim0lem  48369