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Theorem ecopqsi 8714
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 5671 . 2 ((𝐵𝐴𝐶𝐴) → ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴))
2 ecopqsi.1 . . . 4 𝑅 ∈ V
32ecelqsi 8713 . . 3 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅))
4 ecopqsi.2 . . 3 𝑆 = ((𝐴 × 𝐴) / 𝑅)
53, 4eleqtrrdi 2849 . 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 397   = wceq 1542  wcel 2107  Vcvv 3446  cop 4593   × cxp 5632  [cec 8647   / cqs 8648
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651  df-qs 8655
This theorem is referenced by:  brecop  8750  recexsrlem  11040
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