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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 3587 | . . 3 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
2 | df-f 5808 | . . 3 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ran 𝐹)) | |
3 | 1, 2 | mpbiran2 956 | . 2 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹 Fn dom 𝐹) |
4 | eqid 2610 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
5 | df-fn 5807 | . . 3 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹)) | |
6 | 4, 5 | mpbiran2 956 | . 2 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹) |
7 | 3, 6 | bitr2i 264 | 1 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ⊆ wss 3540 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-fn 5807 df-f 5808 |
This theorem is referenced by: nvof1o 6436 rinvf1o 28814 smatrcl 29190 locfinref 29236 fco3 38416 limccog 38687 umgrwwlks2on 41161 |
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