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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 2763 | . . . 4 ⊢ 𝑧 ∈ V | |
2 | vex 2763 | . . . 4 ⊢ 𝑤 ∈ V | |
3 | opeq1 3804 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
4 | 3 | eleq1d 2262 | . . . 4 ⊢ (𝑥 = 𝑧 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
5 | opeq2 3805 | . . . . 5 ⊢ (𝑦 = 𝑤 → 〈𝑧, 𝑦〉 = 〈𝑧, 𝑤〉) | |
6 | 5 | eleq1d 2262 | . . . 4 ⊢ (𝑦 = 𝑤 → (〈𝑧, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑤〉 ∈ 𝐴)) |
7 | 1, 2, 4, 6 | opelopab 4302 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
8 | 7 | gen2 1461 | . 2 ⊢ ∀𝑧∀𝑤(〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
9 | relopab 4788 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} | |
10 | eqrel 4748 | . . 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 424 | . 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 168 | 1 ⊢ (Rel 𝐴 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1362 = wceq 1364 ∈ wcel 2164 〈cop 3621 {copab 4089 Rel wrel 4664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-opab 4091 df-xp 4665 df-rel 4666 |
This theorem is referenced by: opabbi2dv 4811 |
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