MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabid2 Structured version   Visualization version   GIF version

Theorem opabid2 5407
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.
StepHypRef Expression
1 vex 3343 . . . 4 𝑧 ∈ V
2 vex 3343 . . . 4 𝑤 ∈ V
3 opeq1 4553 . . . . 5 (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
43eleq1d 2824 . . . 4 (𝑥 = 𝑧 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5 opeq2 4554 . . . . 5 (𝑦 = 𝑤 → ⟨𝑧, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
65eleq1d 2824 . . . 4 (𝑦 = 𝑤 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴))
71, 2, 4, 6opelopab 5147 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
87gen2 1872 . 2 𝑧𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)
9 relopab 5403 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴}
10 eqrel 5366 . . 3 ((Rel {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ∧ Rel 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴 ↔ ∀𝑧𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)))
119, 10mpan 708 . 2 (Rel 𝐴 → ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴 ↔ ∀𝑧𝑤(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ↔ ⟨𝑧, 𝑤⟩ ∈ 𝐴)))
128, 11mpbiri 248 1 (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630   = wceq 1632  wcel 2139  cop 4327  {copab 4864  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  opabbi2dv  5427
  Copyright terms: Public domain W3C validator