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Mirrors > Home > ILE Home > Th. List > f1ocnv | GIF version |
Description: The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
f1ocnv | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 4997 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | dfrel2 4771 | . . . . . 6 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
3 | fneq1 4987 | . . . . . . 7 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐴)) | |
4 | 3 | biimprd 147 | . . . . . 6 ⊢ (◡◡𝐹 = 𝐹 → (𝐹 Fn 𝐴 → ◡◡𝐹 Fn 𝐴)) |
5 | 2, 4 | sylbi 114 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 Fn 𝐴 → ◡◡𝐹 Fn 𝐴)) |
6 | 1, 5 | mpcom 32 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ◡◡𝐹 Fn 𝐴) |
7 | 6 | anim2i 324 | . . 3 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐹 Fn 𝐴) → (◡𝐹 Fn 𝐵 ∧ ◡◡𝐹 Fn 𝐴)) |
8 | 7 | ancoms 255 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵) → (◡𝐹 Fn 𝐵 ∧ ◡◡𝐹 Fn 𝐴)) |
9 | dff1o4 5134 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) | |
10 | dff1o4 5134 | . 2 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 ↔ (◡𝐹 Fn 𝐵 ∧ ◡◡𝐹 Fn 𝐴)) | |
11 | 8, 9, 10 | 3imtr4i 190 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ◡ccnv 4344 Rel wrel 4350 Fn wfn 4897 –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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: f1ocnvb 5140 f1orescnv 5142 f1imacnv 5143 f1cnv 5150 f1ococnv1 5155 f1oresrab 5329 f1ocnvfv2 5418 f1ocnvdm 5421 f1ocnvfvrneq 5422 fcof1o 5429 isocnv 5451 f1ofveu 5500 ener 6259 en0 6275 en1 6279 ordiso2 6357 cnrecnv 9510 |
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