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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 5626 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴)) | |
2 | ecopqsi.1 | . . . 4 ⊢ 𝑅 ∈ V | |
3 | 2 | ecelqsi 8562 | . . 3 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅)) |
4 | ecopqsi.2 | . . 3 ⊢ 𝑆 = ((𝐴 × 𝐴) / 𝑅) | |
5 | 3, 4 | eleqtrrdi 2850 | . 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 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 〈cop 4567 × cxp 5587 [cec 8496 / cqs 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ec 8500 df-qs 8504 |
This theorem is referenced by: brecop 8599 recexsrlem 10859 |
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