Theorem opabid2 5776

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 3443	. . . 4 ⊢ 𝑧 ∈ V
2		vex 3443	. . . 4 ⊢ 𝑤 ∈ V
3		opeq1 4828	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
4	3	eleq1d 2820	. . . 4 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5		opeq2 4829	. . . . 5 ⊢ (𝑦 = 𝑤 → ⟨𝑧, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
6	5	eleq1d 2820	. . . 4 ⊢ (𝑦 = 𝑤 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴))
7	1, 2, 4, 6	opelopab 5489	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
8	7	gen2 1798	. 2 ⊢ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
9		relopabv 5769	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴}
10		eqrel 5732	. . 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 691	. 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 258	1 ⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ⟨cop 4585  {copab 5159  Rel wrel 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5160  df-xp 5629  df-rel 5630

This theorem is referenced by:  opabbi2dv  5797  tfsconcat0i  43624