Theorem ecopqsi 8764

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 5698	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴))
2		ecopqsi.1	. . . 4 ⊢ 𝑅 ∈ V
3	2	ecelqsi 8763	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅))
4		ecopqsi.2	. . 3 ⊢ 𝑆 = ((𝐴 × 𝐴) / 𝑅)
5	3, 4	eleqtrrdi 2874	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ 𝑆)
6	1, 5	syl 18	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595   × cxp 5659  [cec 8688   / cqs 8689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-qs 8696

This theorem is referenced by:  brecop  8804  recexsrlem  11083