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Mirrors > Home > ILE Home > Th. List > f1f1orn | 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 5093 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
2 | df-f1 4907 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
3 | 2 | simprbi 260 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) |
4 | f1orn 5136 | . 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) | |
5 | 1, 3, 4 | sylanbrc 394 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ◡ccnv 4344 ran crn 4346 Fun wfun 4896 Fn wfn 4897 ⟶wf 4898 –1-1→wf1 4899 –1-1-onto→wf1o 4901 |
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-3an 887 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: f1ores 5141 f1cnv 5150 f1cocnv1 5156 f1ocnvfvrneq 5422 |
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