Theorem ecopqsi 8799

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	⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ 𝑆)

Proof of Theorem ecopqsi

Step	Hyp	Ref	Expression
1		opelxpi 5719	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴))
2		ecopqsi.1	. . . 4 ⊢ 𝑅 ∈ V
3	2	ecelqsi 8798	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅))
4		ecopqsi.2	. . 3 ⊢ 𝑆 = ((𝐴 × 𝐴) / 𝑅)
5	3, 4	eleqtrrdi 2840	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ 𝑆)
6	1, 5	syl 17	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473  ⟨cop 4638   × cxp 5680  [cec 8729   / cqs 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-qs 8737

This theorem is referenced by:  brecop  8835  recexsrlem  11134