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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 2139 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}) | |
2 | snexg 4108 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ V) | |
3 | cnvexg 5076 | . . . 4 ⊢ ({𝑥} ∈ V → ◡{𝑥} ∈ V) | |
4 | uniexg 4361 | . . . 4 ⊢ (◡{𝑥} ∈ V → ∪ ◡{𝑥} ∈ V) | |
5 | 2, 3, 4 | 3syl 17 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ ◡{𝑥} ∈ V) |
6 | 5 | adantl 275 | . 2 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → ∪ ◡{𝑥} ∈ V) |
7 | snexg 4108 | . . . 4 ⊢ (𝑦 ∈ ◡𝐴 → {𝑦} ∈ V) | |
8 | cnvexg 5076 | . . . 4 ⊢ ({𝑦} ∈ V → ◡{𝑦} ∈ V) | |
9 | uniexg 4361 | . . . 4 ⊢ (◡{𝑦} ∈ V → ∪ ◡{𝑦} ∈ V) | |
10 | 7, 8, 9 | 3syl 17 | . . 3 ⊢ (𝑦 ∈ ◡𝐴 → ∪ ◡{𝑦} ∈ V) |
11 | 10 | adantl 275 | . 2 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ ◡𝐴) → ∪ ◡{𝑦} ∈ V) |
12 | cnvf1olem 6121 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) | |
13 | relcnv 4917 | . . . . 5 ⊢ Rel ◡𝐴 | |
14 | simpr 109 | . . . . 5 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) | |
15 | cnvf1olem 6121 | . . . . 5 ⊢ ((Rel ◡𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) | |
16 | 13, 14, 15 | sylancr 410 | . . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) |
17 | dfrel2 4989 | . . . . . . 7 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
18 | eleq2 2203 | . . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴)) | |
19 | 17, 18 | sylbi 120 | . . . . . 6 ⊢ (Rel 𝐴 → (𝑥 ∈ ◡◡𝐴 ↔ 𝑥 ∈ 𝐴)) |
20 | 19 | anbi1d 460 | . . . . 5 ⊢ (Rel 𝐴 → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))) |
21 | 20 | adantr 274 | . . . 4 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → ((𝑥 ∈ ◡◡𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}))) |
22 | 16, 21 | mpbid 146 | . . 3 ⊢ ((Rel 𝐴 ∧ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦})) → (𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥})) |
23 | 12, 22 | impbida 585 | . 2 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = ∪ ◡{𝑥}) ↔ (𝑦 ∈ ◡𝐴 ∧ 𝑥 = ∪ ◡{𝑦}))) |
24 | 1, 6, 11, 23 | f1od 5973 | 1 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ◡{𝑥}):𝐴–1-1-onto→◡𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 {csn 3527 ∪ cuni 3736 ↦ cmpt 3989 ◡ccnv 4538 Rel wrel 4544 –1-1-onto→wf1o 5122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: tposf12 6166 cnven 6702 xpcomf1o 6719 fsumcnv 11213 |
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