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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 5592 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴)) | |
2 | ecopqsi.1 | . . . 4 ⊢ 𝑅 ∈ V | |
3 | 2 | ecelqsi 8353 | . . 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 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 × cxp 5553 [cec 8287 / cqs 8288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ec 8291 df-qs 8295 |
This theorem is referenced by: brecop 8390 recexsrlem 10525 |
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