Theorem opabid2 5811

Description: A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)

Assertion

Ref	Expression
opabid2	⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem opabid2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . 4 ⊢ 𝑧 ∈ V
2		vex 3458	. . . 4 ⊢ 𝑤 ∈ V
3		opeq1 4843	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
4	3	eleq1d 2849	. . . 4 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5		opeq2 4844	. . . . 5 ⊢ (𝑦 = 𝑤 → ⟨𝑧, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
6	5	eleq1d 2849	. . . 4 ⊢ (𝑦 = 𝑤 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴))
7	1, 2, 4, 6	opelopab 5521	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
8	7	gen2 1829	. 2 ⊢ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
9		relopabv 5804	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴}
10		eqrel 5765	. . 3 ⊢ ((Rel {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ∧ Rel 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴 ↔ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)))
11	9, 10	mpan 703	. 2 ⊢ (Rel 𝐴 → ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴 ↔ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)))
12	8, 11	mpbiri 261	1 ⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ⟨cop 4838  {copab 5170  Rel wrel 5660

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 5252  ax-pr 5395

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-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 4839  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by:  opabbi2dv  5832  tfsconcat0i  44537