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Mirrors > Home > ILE Home > Th. List > opabid2 | GIF version |
Description: A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.) |
Ref | Expression |
---|---|
opabid2 | ⊢ (Rel 𝐴 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2604 | . . . 4 ⊢ 𝑧 ∈ V | |
2 | vex 2604 | . . . 4 ⊢ 𝑤 ∈ V | |
3 | opeq1 3570 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
4 | 3 | eleq1d 2147 | . . . 4 ⊢ (𝑥 = 𝑧 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
5 | opeq2 3571 | . . . . 5 ⊢ (𝑦 = 𝑤 → 〈𝑧, 𝑦〉 = 〈𝑧, 𝑤〉) | |
6 | 5 | eleq1d 2147 | . . . 4 ⊢ (𝑦 = 𝑤 → (〈𝑧, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑤〉 ∈ 𝐴)) |
7 | 1, 2, 4, 6 | opelopab 4026 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
8 | 7 | gen2 1379 | . 2 ⊢ ∀𝑧∀𝑤(〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
9 | relopab 4482 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} | |
10 | eqrel 4447 | . . 3 ⊢ ((Rel {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ∧ Rel 𝐴) → ({〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴 ↔ ∀𝑧∀𝑤(〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴))) | |
11 | 9, 10 | mpan 414 | . 2 ⊢ (Rel 𝐴 → ({〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴 ↔ ∀𝑧∀𝑤(〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴))) |
12 | 8, 11 | mpbiri 166 | 1 ⊢ (Rel 𝐴 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 = wceq 1284 ∈ wcel 1433 〈cop 3401 {copab 3838 Rel wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 df-rel 4370 |
This theorem is referenced by: opabbi2dv 4503 |
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