Theorem opabid2 5173

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 3176	. . . 4 ⊢ 𝑧 ∈ V
2		vex 3176	. . . 4 ⊢ 𝑤 ∈ V
3		opeq1 4340	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
4	3	eleq1d 2672	. . . 4 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5		opeq2 4341	. . . . 5 ⊢ (𝑦 = 𝑤 → ⟨𝑧, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
6	5	eleq1d 2672	. . . 4 ⊢ (𝑦 = 𝑤 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴))
7	1, 2, 4, 6	opelopab 4922	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
8	7	gen2 1714	. 2 ⊢ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
9		relopab 5169	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴}
10		eqrel 5132	. . 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 702	. 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 247	1 ⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  {copab 4642  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045

This theorem is referenced by:  opabbi2dv  5193