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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 6820 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
| 2 | forn 6823 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
| 4 | ffn 6736 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
| 5 | df-fo 6567 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
| 6 | 5 | biimpri 228 | . . 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 209 | 1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ran crn 5686 Fn wfn 6556 ⟶wf 6557 –onto→wfo 6559 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ss 3968 df-f 6565 df-fo 6567 |
| This theorem is referenced by: focofo 6833 foconst 6835 dff1o5 6857 dffo3 7122 dffo4 7123 exfo 7125 dffo3f 7126 fo1stres 8040 fo2ndres 8041 fo2ndf 8146 cantnf 9733 hsmexlem2 10467 setcepi 18133 odf1o1 19590 efgsfo 19757 pjfo 21735 xrhmeo 24977 grpofo 30518 cnpconn 35235 lnmepi 43097 imasetpreimafvbijlemfo 47392 fargshiftfo 47429 |
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