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Mirrors > Home > ILE Home > Th. List > ecopqsi | 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 4692 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴)) | |
2 | ecopqsi.1 | . . . 4 ⊢ 𝑅 ∈ V | |
3 | 2 | ecelqsi 6645 | . . 3 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅)) |
4 | ecopqsi.2 | . . 3 ⊢ 𝑆 = ((𝐴 × 𝐴) / 𝑅) | |
5 | 3, 4 | eleqtrrdi 2287 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ 𝑆) |
6 | 1, 5 | syl 14 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → [〈𝐵, 𝐶〉]𝑅 ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 Vcvv 2760 〈cop 3622 × cxp 4658 [cec 6587 / cqs 6588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-xp 4666 df-cnv 4668 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-ec 6591 df-qs 6595 |
This theorem is referenced by: brecop 6681 recexgt0sr 7835 ltpsrprg 7865 |
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