![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fdmrn | Structured version Visualization version GIF version |
Description: A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.) |
Ref | Expression |
---|---|
fdmrn | ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4018 | . . 3 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
2 | df-f 6567 | . . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹)) | |
3 | 1, 2 | mpbiran2 710 | . 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹) |
4 | eqid 2735 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
5 | df-fn 6566 | . . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹)) | |
6 | 4, 5 | mpbiran2 710 | . 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹) |
7 | 3, 6 | bitr2i 276 | 1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ⊆ wss 3963 dom cdm 5689 ran crn 5690 Fun wfun 6557 Fn wfn 6558 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-ss 3980 df-fn 6566 df-f 6567 |
This theorem is referenced by: funcofd 6769 nvof1o 7300 umgrwwlks2on 29987 rinvf1o 32647 smatrcl 33757 locfinref 33802 lfuhgr 35102 limccog 45576 funfocofob 47028 isuspgrim0lem 47809 grimuhgr 47816 |
Copyright terms: Public domain | W3C validator |