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Mirrors > Home > ILE Home > Th. List > dff1o5 | GIF version |
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
dff1o5 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1o 5100 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
2 | f1f 5298 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | biantrurd 303 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))) |
4 | dffo2 5319 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) | |
5 | 3, 4 | syl6rbbr 198 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹:𝐴–onto→𝐵 ↔ ran 𝐹 = 𝐵)) |
6 | 5 | pm5.32i 449 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) |
7 | 1, 6 | bitri 183 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1316 ran crn 4510 ⟶wf 5089 –1-1→wf1 5090 –onto→wfo 5091 –1-1-onto→wf1o 5092 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-in 3047 df-ss 3054 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 |
This theorem is referenced by: f1orescnv 5351 f1finf1o 6803 djuinr 6916 eninl 6950 eninr 6951 frec2uzf1od 10147 ennnfonelemex 11854 ennnfonelemen 11861 pwf1oexmid 13121 |
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