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Mirrors > Home > MPE Home > Th. List > ecopqsi | Structured version Visualization version GIF version |
Description: "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.) |
Ref | Expression |
---|---|
ecopqsi.1 | ⊢ 𝑅 ∈ V |
ecopqsi.2 | ⊢ 𝑆 = ((𝐴 × 𝐴) / 𝑅) |
Ref | Expression |
---|---|
ecopqsi | ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5587 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴)) | |
2 | ecopqsi.1 | . . . 4 ⊢ 𝑅 ∈ V | |
3 | 2 | ecelqsi 8347 | . . 3 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅)) |
4 | ecopqsi.2 | . . 3 ⊢ 𝑆 = ((𝐴 × 𝐴) / 𝑅) | |
5 | 3, 4 | eleqtrrdi 2924 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ 𝑆) |
6 | 1, 5 | syl 17 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 〈cop 4567 × cxp 5548 [cec 8281 / cqs 8282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ec 8285 df-qs 8289 |
This theorem is referenced by: brecop 8384 recexsrlem 10519 |
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