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Theorem brabd0 35830
Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
Hypotheses
Ref Expression
brabd0.x (𝜑 → ∀𝑥𝜑)
brabd0.y (𝜑 → ∀𝑦𝜑)
brabd0.xch (𝜑 → Ⅎ𝑥𝜒)
brabd0.ych (𝜑 → Ⅎ𝑦𝜒)
brabd0.exa (𝜑𝐴𝑈)
brabd0.exb (𝜑𝐵𝑉)
brabd0.def (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
brabd0.is ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
brabd0 (𝜑 → (𝐴𝑅𝐵𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem brabd0
StepHypRef Expression
1 df-br 5142 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 brabd0.def . . . 4 (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
32eleq2d 2818 . . 3 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
41, 3bitrid 282 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
5 brabd0.x . . 3 (𝜑 → ∀𝑥𝜑)
6 brabd0.y . . 3 (𝜑 → ∀𝑦𝜑)
7 brabd0.xch . . 3 (𝜑 → Ⅎ𝑥𝜒)
8 brabd0.ych . . 3 (𝜑 → Ⅎ𝑦𝜒)
9 brabd0.exa . . 3 (𝜑𝐴𝑈)
10 brabd0.exb . . 3 (𝜑𝐵𝑉)
11 brabd0.is . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
125, 6, 7, 8, 9, 10, 11opelopabd 35824 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
134, 12bitrd 278 1 (𝜑 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wnf 1785  wcel 2106  cop 4628   class class class wbr 5141  {copab 5203
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204
This theorem is referenced by:  brabd  35831
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