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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 2715 | . . . 4 ⊢ 𝑧 ∈ V | |
2 | vex 2715 | . . . 4 ⊢ 𝑤 ∈ V | |
3 | opeq1 3743 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
4 | 3 | eleq1d 2226 | . . . 4 ⊢ (𝑥 = 𝑧 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
5 | opeq2 3744 | . . . . 5 ⊢ (𝑦 = 𝑤 → 〈𝑧, 𝑦〉 = 〈𝑧, 𝑤〉) | |
6 | 5 | eleq1d 2226 | . . . 4 ⊢ (𝑦 = 𝑤 → (〈𝑧, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑤〉 ∈ 𝐴)) |
7 | 1, 2, 4, 6 | opelopab 4234 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
8 | 7 | gen2 1430 | . 2 ⊢ ∀𝑧∀𝑤(〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
9 | relopab 4716 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} | |
10 | eqrel 4678 | . . 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 421 | . 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 167 | 1 ⊢ (Rel 𝐴 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1333 = wceq 1335 ∈ wcel 2128 〈cop 3564 {copab 4027 Rel wrel 4594 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-opab 4029 df-xp 4595 df-rel 4596 |
This theorem is referenced by: opabbi2dv 4738 |
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