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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 6355 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
2 | f1f 6568 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | biantrurd 533 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))) |
4 | dffo2 6587 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) | |
5 | 3, 4 | syl6rbbr 291 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹:𝐴–onto→𝐵 ↔ ran 𝐹 = 𝐵)) |
6 | 5 | pm5.32i 575 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) |
7 | 1, 6 | bitri 276 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ran crn 5549 ⟶wf 6344 –1-1→wf1 6345 –onto→wfo 6346 –1-1-onto→wf1o 6347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-in 3940 df-ss 3949 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 |
This theorem is referenced by: f1orescnv 6623 domdifsn 8588 sucdom2 8702 ackbij1 9648 ackbij2 9653 fin4en1 9719 om2uzf1oi 13309 s4f1o 14268 fvcosymgeq 18486 indlcim 20912 2lgslem1b 25895 ausgrusgrb 26877 usgrexmpledg 26971 cdleme50f1o 37562 diaf1oN 38146 pwssplit4 39567 meadjiunlem 42624 |
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