Theorem brab2dd 48691

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)})
brab2dd.2	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
brab2dd.3	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))

Assertion

Ref	Expression
brab2dd	⊢ (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)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brab2dd

Step	Hyp	Ref	Expression
1		df-br 5152	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		brab2dd.1	. . . . 5 ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)})
3	2	eleq2d 2827	. . . 4 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)}))
4	1, 3	bitrid 283	. . 3 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)}))
5		elopab 5541	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)))
6	4, 5	bitrdi 287	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓))))
7		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓))) → 𝜑)
8		eqcom 2744	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩)
9		vex 3485	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
10		vex 3485	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
11	9, 10	opth 5490	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
12	8, 11	sylbb1 237	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
13	12	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓))) → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
14		simprrl 781	. . . . . . 7 ⊢ ((𝜑 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓))) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
15		brab2dd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
16	15	biimpa 476	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
17	7, 13, 14, 16	syl21anc 838	. . . . . 6 ⊢ ((𝜑 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓))) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
18	17	ex 412	. . . . 5 ⊢ (𝜑 → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
19	18	exlimdvv 1934	. . . 4 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
20	19	imp 406	. . 3 ⊢ ((𝜑 ∧ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓))) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
21		simprl 771	. . 3 ⊢ ((𝜑 ∧ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
22		simprl 771	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑈)
23		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
24		brab2dd.2	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
25	15, 24	anbi12d 632	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
26	25	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
27	22, 23, 26	copsex2dv 5508	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
28	20, 21, 27	bibiad 840	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
29	6, 28	bitrd 279	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2108  ⟨cop 4640   class class class wbr 5151  {copab 5213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152  df-opab 5214

This theorem is referenced by:  brab2ddw  48692  brab2ddw2  48693