Theorem fdmrn 6365

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 3874	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6190	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 698	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2773	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6189	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 698	. 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7	3, 6	bitr2i 268	1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

Syntax hints:   ↔ wb 198   = wceq 1508   ⊆ wss 3824  dom cdm 5404  ran crn 5405  Fun wfun 6180   Fn wfn 6181  ⟶wf 6182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-in 3831  df-ss 3838  df-fn 6189  df-f 6190

This theorem is referenced by:  nvof1o  6861  umgrwwlks2on  27479  rinvf1o  30156  smatrcl  30736  locfinref  30782  fco3  40947  limccog  41362