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Mirrors > Home > MPE Home > Th. List > dffo2 | Structured version Visualization version GIF version |
Description: Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.) |
Ref | Expression |
---|---|
dffo2 | ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fof 6583 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | forn 6586 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
3 | 1, 2 | jca 512 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
4 | ffn 6507 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
5 | df-fo 6354 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
6 | 5 | biimpri 229 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵) |
7 | 4, 6 | sylan 580 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵) |
8 | 3, 7 | impbii 210 | 1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ran crn 5549 Fn wfn 6343 ⟶wf 6344 –onto→wfo 6346 |
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-fo 6354 |
This theorem is referenced by: foco 6595 foconst 6596 dff1o5 6617 dffo3 6860 dffo4 6861 exfo 6863 fo1stres 7704 fo2ndres 7705 fo2ndf 7806 cantnf 9144 hsmexlem2 9837 setcepi 17336 odf1o1 18626 efgsfo 18794 pjfo 20787 xrhmeo 23477 grpofo 28203 cnpconn 32374 lnmepi 39563 dffo3f 41314 imasetpreimafvbijlemfo 43442 fargshiftfo 43479 |
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