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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 6028 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | forn 6031 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
3 | 1, 2 | jca 553 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
4 | ffn 5958 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
5 | df-fo 5810 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
6 | 5 | biimpri 217 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵) |
7 | 4, 6 | sylan 487 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵) |
8 | 3, 7 | impbii 198 | 1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ran crn 5039 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-f 5808 df-fo 5810 |
This theorem is referenced by: foco 6038 foconst 6039 dff1o5 6059 dffo3 6282 dffo4 6283 exfo 6285 fo1stres 7083 fo2ndres 7084 fo2ndf 7171 cantnf 8473 hsmexlem2 9132 setcepi 16561 odf1o1 17810 efgsfo 17975 pjfo 19878 xrhmeo 22553 fargshiftfo 26166 grpofo 26737 cnpcon 30466 lnmepi 36673 dffo3f 38359 |
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