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| Mirrors > Home > MPE Home > Th. List > f1f1orn | Structured version Visualization version GIF version | ||
| Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.) |
| Ref | Expression |
|---|---|
| f1f1orn | ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1fn 6805 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
| 2 | df-f1 6566 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
| 3 | 2 | simprbi 496 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) |
| 4 | f1orn 6858 | . 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) | |
| 5 | 1, 3, 4 | sylanbrc 583 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) |
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