Theorem fdmrn 6746

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 4003	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2		df-f 6544	. . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹))
3	1, 2	mpbiran2 708	. 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹)
4		eqid 2732	. . 3 ⊢ dom 𝐹 = dom 𝐹
5		df-fn 6543	. . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
6	4, 5	mpbiran2 708	. 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7	3, 6	bitr2i 275	1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)

Syntax hints:   ↔ wb 205   = wceq 1541   ⊆ wss 3947  dom cdm 5675  ran crn 5676  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-fn 6543  df-f 6544

This theorem is referenced by:  funcofd  6747  fco3OLD  6748  nvof1o  7274  umgrwwlks2on  29200  rinvf1o  31841  smatrcl  32764  locfinref  32809  lfuhgr  34096  limccog  44322  funfocofob  45772