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Mirrors > Home > ILE Home > Th. List > dffn3 | GIF version |
Description: A function maps to its range. (Contributed by NM, 1-Sep-1999.) |
Ref | Expression |
---|---|
dffn3 | ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 2964 | . . 3 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
2 | 1 | biantru 286 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹)) |
3 | df-f 4906 | . 2 ⊢ (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹)) | |
4 | 2, 3 | bitr4i 176 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ⊆ wss 2917 ran crn 4346 Fn wfn 4897 ⟶wf 4898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 df-f 4906 |
This theorem is referenced by: fsn2 5337 fo2ndf 5848 |
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