Theorem opabid2 4867

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 2806	. . . 4 ⊢ 𝑧 ∈ V
2		vex 2806	. . . 4 ⊢ 𝑤 ∈ V
3		opeq1 3867	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
4	3	eleq1d 2300	. . . 4 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5		opeq2 3868	. . . . 5 ⊢ (𝑦 = 𝑤 → ⟨𝑧, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
6	5	eleq1d 2300	. . . 4 ⊢ (𝑦 = 𝑤 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴))
7	1, 2, 4, 6	opelopab 4372	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
8	7	gen2 1499	. 2 ⊢ ∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
9		relopab 4862	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴}
10		eqrel 4821	. . 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 424	. 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 168	1 ⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2202  ⟨cop 3676  {copab 4154  Rel wrel 4736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-xp 4737  df-rel 4738

This theorem is referenced by:  opabbi2dv  4885