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Mirrors > Home > ILE Home > Th. List > f1ocnvb | GIF version |
Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.) |
Ref | Expression |
---|---|
f1ocnvb | ⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 5334 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1ocnv 5334 | . . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡◡𝐹:𝐴–1-1-onto→𝐵) | |
3 | dfrel2 4945 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
4 | f1oeq1 5312 | . . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | |
5 | 3, 4 | sylbi 120 | . . 3 ⊢ (Rel 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) |
6 | 2, 5 | syl5ib 153 | . 2 ⊢ (Rel 𝐹 → (◡𝐹:𝐵–1-1-onto→𝐴 → 𝐹:𝐴–1-1-onto→𝐵)) |
7 | 1, 6 | impbid2 142 | 1 ⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1312 ◡ccnv 4496 Rel wrel 4502 –1-1-onto→wf1o 5078 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-br 3894 df-opab 3948 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 |
This theorem is referenced by: (None) |
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