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Mirrors > Home > MPE Home > Th. List > opabid2 | Structured version Visualization version 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 3495 | . . . 4 ⊢ 𝑧 ∈ V | |
2 | vex 3495 | . . . 4 ⊢ 𝑤 ∈ V | |
3 | opeq1 4795 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
4 | 3 | eleq1d 2894 | . . . 4 ⊢ (𝑥 = 𝑧 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
5 | opeq2 4796 | . . . . 5 ⊢ (𝑦 = 𝑤 → 〈𝑧, 𝑦〉 = 〈𝑧, 𝑤〉) | |
6 | 5 | eleq1d 2894 | . . . 4 ⊢ (𝑦 = 𝑤 → (〈𝑧, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑤〉 ∈ 𝐴)) |
7 | 1, 2, 4, 6 | opelopab 5420 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
8 | 7 | gen2 1788 | . 2 ⊢ ∀𝑧∀𝑤(〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
9 | relopab 5689 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} | |
10 | eqrel 5651 | . . 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 686 | . 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 259 | 1 ⊢ (Rel 𝐴 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 = wceq 1528 ∈ wcel 2105 〈cop 4563 {copab 5119 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: opabbi2dv 5713 |
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