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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 4000 | . . 3 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
2 | df-f 6536 | . . 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 2731 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
5 | df-fn 6535 | . . 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) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ⊆ wss 3944 dom cdm 5669 ran crn 5670 Fun wfun 6526 Fn wfn 6527 ⟶wf 6528 |
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 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3951 df-ss 3961 df-fn 6535 df-f 6536 |
This theorem is referenced by: funcofd 6737 fco3OLD 6738 nvof1o 7262 umgrwwlks2on 29076 rinvf1o 31722 smatrcl 32607 locfinref 32652 lfuhgr 33939 limccog 44109 funfocofob 45558 |
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