Theorem brabd0 37396

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	⊢ (𝜑 → 𝐴 ∈ 𝑈)
brabd0.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
brabd0.def	⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
brabd0.is	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
brabd0	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))

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

Step	Hyp	Ref	Expression
1		df-br 5101	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		brabd0.def	. . . 4 ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
3	2	eleq2d 2823	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
4	1, 3	bitrid 283	. 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 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
10		brabd0.exb	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
11		brabd0.is	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
12	5, 6, 7, 8, 9, 10, 11	opelopabd 37390	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
13	4, 12	bitrd 279	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  {copab 5162

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

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-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163

This theorem is referenced by:  brabd  37397