Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opelopabd Structured version   Visualization version   GIF version

Theorem opelopabd 34517
 Description: Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
Hypotheses
Ref Expression
opelopabd.xph (𝜑 → ∀𝑥𝜑)
opelopabd.yph (𝜑 → ∀𝑦𝜑)
opelopabd.xch (𝜑 → Ⅎ𝑥𝜒)
opelopabd.ych (𝜑 → Ⅎ𝑦𝜒)
opelopabd.exa (𝜑𝐴𝑈)
opelopabd.exb (𝜑𝐵𝑉)
opelopabd.is ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
opelopabd (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem opelopabd
StepHypRef Expression
1 elopab 5391 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
2 opelopabd.xph . . 3 (𝜑 → ∀𝑥𝜑)
3 opelopabd.yph . . 3 (𝜑 → ∀𝑦𝜑)
4 opelopabd.xch . . 3 (𝜑 → Ⅎ𝑥𝜒)
5 opelopabd.ych . . 3 (𝜑 → Ⅎ𝑦𝜒)
6 opelopabd.exa . . 3 (𝜑𝐴𝑈)
7 opelopabd.exb . . 3 (𝜑𝐵𝑉)
8 opelopabd.is . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
92, 3, 4, 5, 6, 7, 8copsex2d 34515 . 2 (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
101, 9syl5bb 286 1 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2114  ⟨cop 4545  {copab 5104 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-opab 5105 This theorem is referenced by:  brabd0  34523
 Copyright terms: Public domain W3C validator