Theorem ecopqsi 8829

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 5695	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴))
2		ecopqsi.1	. . . 4 ⊢ 𝑅 ∈ V
3	2	ecelqsi 8828	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ ((𝐴 × 𝐴) / 𝑅))
4		ecopqsi.2	. . 3 ⊢ 𝑆 = ((𝐴 × 𝐴) / 𝑅)
5	3, 4	eleqtrrdi 2875	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ 𝑆)
6	1, 5	syl 18	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → [⟨𝐵, 𝐶⟩]𝑅 ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840   × cxp 5656  [cec 8753   / cqs 8754

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-qs 8761

This theorem is used by:  brecop  8869  recexsrlem  11227