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Mirrors > Home > ILE Home > Th. List > cnvf1o | GIF version |
Description: Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
cnvf1o | ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2040 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) | |
2 | snexg 3936 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ V) | |
3 | cnvexg 4855 | . . . 4 ⊢ ({𝑥} ∈ V → ◡{𝑥} ∈ V) | |
4 | uniexg 4175 | . . . 4 ⊢ (◡{𝑥} ∈ V → ∪ ◡{𝑥} ∈ V) | |
5 | 2, 3, 4 | 3syl 17 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ ◡{𝑥} ∈ V) |
6 | 5 | adantl 262 | . 2 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∪ ◡{𝑥} ∈ V) |
7 | snexg 3936 | . . . 4 ⊢ (𝑦 ∈ ◡𝐴 → {𝑦} ∈ V) | |
8 | cnvexg 4855 | . . . 4 ⊢ ({𝑦} ∈ V → ◡{𝑦} ∈ V) | |
9 | uniexg 4175 | . . . 4 ⊢ (◡{𝑦} ∈ V → ∪ ◡{𝑦} ∈ V) | |
10 | 7, 8, 9 | 3syl 17 | . . 3 ⊢ (𝑦 ∈ ◡𝐴 → ∪ ◡{𝑦} ∈ V) |
11 | 10 | adantl 262 | . 2 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ ◡𝐴) → ∪ ◡{𝑦} ∈ V) |
12 | cnvf1olem 5845 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) | |
13 | relcnv 4703 | . . . . 5 ⊢ Rel ◡𝐴 | |
14 | simpr 103 | . . . . 5 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) | |
15 | cnvf1olem 5845 | . . . . 5 ⊢ ((Rel ◡𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) | |
16 | 13, 14, 15 | sylancr 393 | . . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) |
17 | dfrel2 4771 | . . . . . . 7 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
18 | eleq2 2101 | . . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴)) | |
19 | 17, 18 | sylbi 114 | . . . . . 6 ⊢ (Rel 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴)) |
20 | 19 | anbi1d 438 | . . . . 5 ⊢ (Rel 𝐴 → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))) |
21 | 20 | adantr 261 | . . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))) |
22 | 16, 21 | mpbid 135 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) |
23 | 12, 22 | impbida 528 | . 2 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))) |
24 | 1, 6, 11, 23 | f1od 5703 | 1 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {csn 3375 ∪ cuni 3580 ↦ cmpt 3818 ◡ccnv 4344 Rel wrel 4350 –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-13 1404 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 ax-un 4170 |
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-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: tposf12 5884 cnven 6288 xpcomf1o 6299 |
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