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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 3400 | . . . 4 ⊢ 𝑧 ∈ V | |
2 | vex 3400 | . . . 4 ⊢ 𝑤 ∈ V | |
3 | opeq1 4636 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
4 | 3 | eleq1d 2843 | . . . 4 ⊢ (𝑥 = 𝑧 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
5 | opeq2 4637 | . . . . 5 ⊢ (𝑦 = 𝑤 → 〈𝑧, 𝑦〉 = 〈𝑧, 𝑤〉) | |
6 | 5 | eleq1d 2843 | . . . 4 ⊢ (𝑦 = 𝑤 → (〈𝑧, 𝑦〉 ∈ 𝐴 ↔ 〈𝑧, 𝑤〉 ∈ 𝐴)) |
7 | 1, 2, 4, 6 | opelopab 5234 | . . 3 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
8 | 7 | gen2 1840 | . 2 ⊢ ∀𝑧∀𝑤(〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} ↔ 〈𝑧, 𝑤〉 ∈ 𝐴) |
9 | relopab 5493 | . . 3 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} | |
10 | eqrel 5456 | . . 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 680 | . 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 250 | 1 ⊢ (Rel 𝐴 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1599 = wceq 1601 ∈ wcel 2106 〈cop 4403 {copab 4948 Rel wrel 5360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-opab 4949 df-xp 5361 df-rel 5362 |
This theorem is referenced by: opabbi2dv 5517 |
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