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Mirrors > Home > ILE Home > Th. List > dffo2 | 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 5106 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | forn 5109 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
3 | 1, 2 | jca 290 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
4 | ffn 5046 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
5 | df-fo 4908 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
6 | 5 | biimpri 124 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵) |
7 | 4, 6 | sylan 267 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵) |
8 | 3, 7 | impbii 117 | 1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ran crn 4346 Fn wfn 4897 ⟶wf 4898 –onto→wfo 4900 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 df-f 4906 df-fo 4908 |
This theorem is referenced by: foco 5116 dff1o5 5135 dffo3 5314 dffo4 5315 fo1stresm 5788 fo2ndresm 5789 fo2ndf 5848 1fv 8996 |
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