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Mirrors > Home > ILE Home > Th. List > f1ofo | GIF version |
Description: A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.) |
Ref | Expression |
---|---|
f1ofo | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o3 5132 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹)) | |
2 | 1 | simplbi 259 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ◡ccnv 4344 Fun wfun 4896 –onto→wfo 4900 –1-1-onto→wf1o 4901 |
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-3an 887 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-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: f1imacnv 5143 f1ococnv2 5153 fo00 5162 isoini 5457 isoselem 5459 f1opw2 5706 f1dmex 5743 bren 6228 f1oeng 6237 en1 6279 phplem4 6318 phplem4on 6329 dif1en 6337 ordiso2 6357 1fv 8996 |
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