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Theorem ecopqsi 8555
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 5627 . 2 ((𝐵𝐴𝐶𝐴) → ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴))
2 ecopqsi.1 . . . 4 𝑅 ∈ V
32ecelqsi 8554 . . 3 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅))
4 ecopqsi.2 . . 3 𝑆 = ((𝐴 × 𝐴) / 𝑅)
53, 4eleqtrrdi 2852 . 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 396   = wceq 1542  wcel 2110  Vcvv 3431  cop 4573   × cxp 5588  [cec 8488   / cqs 8489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7583
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5596  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ec 8492  df-qs 8496
This theorem is referenced by:  brecop  8591  recexsrlem  10870
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