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Theorem ecopqsi 7756
Description: "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
Hypotheses
Ref Expression
ecopqsi.1 𝑅 ∈ V
ecopqsi.2 𝑆 = ((𝐴 × 𝐴) / 𝑅)
Assertion
Ref Expression
ecopqsi ((𝐵𝐴𝐶𝐴) → [⟨𝐵, 𝐶⟩]𝑅𝑆)

Proof of Theorem ecopqsi
StepHypRef Expression
1 opelxpi 5113 . 2 ((𝐵𝐴𝐶𝐴) → ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴))
2 ecopqsi.1 . . . 4 𝑅 ∈ V
32ecelqsi 7755 . . 3 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅))
4 ecopqsi.2 . . 3 𝑆 = ((𝐴 × 𝐴) / 𝑅)
53, 4syl6eleqr 2709 . 2 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅𝑆)
61, 5syl 17 1 ((𝐵𝐴𝐶𝐴) → [⟨𝐵, 𝐶⟩]𝑅𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  cop 4159   × cxp 5077  [cec 7692   / cqs 7693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ec 7696  df-qs 7700
This theorem is referenced by:  brecop  7792  recexsrlem  9876
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