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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 3018 | . . 3 ⊢ ran 𝐹 ⊆ ran 𝐹 | |
2 | 1 | biantru 296 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹)) |
3 | df-f 4926 | . 2 ⊢ (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹)) | |
4 | 2, 3 | bitr4i 185 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ⊆ wss 2973 ran crn 4364 Fn wfn 4917 ⟶wf 4918 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-in 2979 df-ss 2986 df-f 4926 |
This theorem is referenced by: fsn2 5358 fo2ndf 5868 |
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