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Mirrors > Home > MPE Home > Th. List > dff1o5 | Structured version Visualization version 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 6570 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
2 | dffo2 6825 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) | |
3 | f1f 6805 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
4 | 3 | biantrurd 532 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))) |
5 | 2, 4 | bitr4id 290 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹:𝐴–onto→𝐵 ↔ ran 𝐹 = 𝐵)) |
6 | 5 | pm5.32i 574 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) |
7 | 1, 6 | bitri 275 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ran crn 5690 ⟶wf 6559 –1-1→wf1 6560 –onto→wfo 6561 –1-1-onto→wf1o 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-ss 3980 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: f1orescnv 6864 f1ounsn 7292 domdifsn 9093 sucdom2OLD 9121 sucdom2 9241 ackbij1 10275 ackbij2 10280 fin4en1 10347 om2uzf1oi 13991 s4f1o 14954 fvcosymgeq 19462 indlcim 21878 2lgslem1b 27451 ausgrusgrb 29197 usgrexmpledg 29294 wrdpmtrlast 33096 cdleme50f1o 40529 diaf1oN 41113 aks6d1c2 42112 pwssplit4 43078 cantnf2 43315 meadjiunlem 46421 |
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