Theorem brabd 37462

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

Assertion

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

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem brabd

Step	Hyp	Ref	Expression
1		ax-5 1912	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax-5 1912	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		nfvd 1917	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4		nfvd 1917	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜒)
5		brabd.exa	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
6		brabd.exb	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		brabd.def	. 2 ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
8		brabd.is	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
9	1, 2, 3, 4, 5, 6, 7, 8	brabd0 37461	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5085  {copab 5147

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 2708  ax-sep 5231  ax-pr 5375

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148

This theorem is referenced by:  bj-imdirval3  37498