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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 5423 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
2 | df-f1 5221 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
3 | 2 | simprbi 275 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) |
4 | f1orn 5471 | . 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) | |
5 | 1, 3, 4 | sylanbrc 417 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ◡ccnv 4625 ran crn 4627 Fun wfun 5210 Fn wfn 5211 ⟶wf 5212 –1-1→wf1 5213 –1-1-onto→wf1o 5215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3135 df-ss 3142 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 |
This theorem is referenced by: f1ores 5476 f1cnv 5485 f1cocnv1 5491 f1ocnvfvrneq 5782 ssenen 6850 f1dmvrnfibi 6942 cc2lem 7264 xpsff1o2 12764 |
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