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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 6757 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | forn 6760 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
3 | 1, 2 | jca 513 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
4 | ffn 6669 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
5 | df-fo 6503 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
6 | 5 | biimpri 227 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵) |
7 | 4, 6 | sylan 581 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴–onto→𝐵) |
8 | 3, 7 | impbii 208 | 1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ran crn 5635 Fn wfn 6492 ⟶wf 6493 –onto→wfo 6495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3448 df-in 3918 df-ss 3928 df-f 6501 df-fo 6503 |
This theorem is referenced by: focofo 6770 foconst 6772 dff1o5 6794 dffo3 7053 dffo4 7054 exfo 7056 fo1stres 7948 fo2ndres 7949 fo2ndf 8054 cantnf 9630 hsmexlem2 10364 setcepi 17975 odf1o1 19355 efgsfo 19522 pjfo 21124 xrhmeo 24312 grpofo 29444 cnpconn 33827 lnmepi 41415 dffo3f 43405 imasetpreimafvbijlemfo 45604 fargshiftfo 45641 |
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