Theorem brab2ddw2 49641

Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by Zhi Wang, 24-Sep-2025.)

Hypotheses

Ref	Expression
brab2dd.1	⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)})
brab2ddw.2	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
brab2ddw.3	⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒))
brab2ddw2.4	⊢ (𝑥 = 𝐴 → 𝐶 = 𝑈)
brab2ddw2.5	⊢ (𝑦 = 𝐵 → 𝐷 = 𝑉)

Assertion

Ref	Expression
brab2ddw2	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))

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

Allowed substitution hints:   𝜓(𝑥, 𝑦)   𝜃(𝑥, 𝑦)   𝐶(𝑥, 𝑦)   𝐷(𝑥, 𝑦)   𝑅(𝑥, 𝑦)

Proof of Theorem brab2ddw2

Step	Hyp	Ref	Expression
1		brab2dd.1	. 2 ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)})
2		brab2ddw.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
3		brab2ddw.3	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒))
4	2, 3	sylan9bb 519	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
5	4	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
6		id 23	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7		brab2ddw2.4	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝑈)
8	6, 7	eleq12d 2859	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝑈))
9		id 23	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
10		brab2ddw2.5	. . . . 5 ⊢ (𝑦 = 𝐵 → 𝐷 = 𝑉)
11	9, 10	eleq12d 2859	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝑉))
12	8, 11	bi2anan9 650	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
13	12	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
14	1, 5, 13	brab2dd 49639	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   class class class wbr 5111  {copab 5175

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176

This theorem is used by: (None)