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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 4006 | . . 3 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
| 2 | df-f 6565 | . . 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 2737 | . . 3 ⊢ dom 𝐹 = dom 𝐹 | |
| 5 | df-fn 6564 | . . 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 1540 ⊆ wss 3951 dom cdm 5685 ran crn 5686 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ss 3968 df-fn 6564 df-f 6565 |
| This theorem is referenced by: funcofd 6768 nvof1o 7300 umgrwwlks2on 29977 rinvf1o 32640 smatrcl 33795 locfinref 33840 lfuhgr 35123 limccog 45635 funfocofob 47090 isuspgrim0lem 47871 grimuhgr 47878 |
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