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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 5389 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
2 | df-f1 5187 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
3 | 2 | simprbi 273 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) |
4 | f1orn 5436 | . 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) | |
5 | 1, 3, 4 | sylanbrc 414 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ◡ccnv 4597 ran crn 4599 Fun wfun 5176 Fn wfn 5177 ⟶wf 5178 –1-1→wf1 5179 –1-1-onto→wf1o 5181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-in 3117 df-ss 3124 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 |
This theorem is referenced by: f1ores 5441 f1cnv 5450 f1cocnv1 5456 f1ocnvfvrneq 5744 ssenen 6808 f1dmvrnfibi 6900 cc2lem 7198 |
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