Theorem brab2d 5512

Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
brab2d.1	⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)})
brab2d.2	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
brab2d	⊢ (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)   𝑅(𝑥, 𝑦)

Proof of Theorem brab2d

Step	Hyp	Ref	Expression
1		df-br 5104	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		brab2d.1	. . . . 5 ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)})
3	2	eleq2d 2847	. . . 4 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)}))
4	1, 3	bitrid 286	. . 3 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)}))
5		elopab 5501	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)))
6	4, 5	bitrdi 290	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓))))
7		eqcom 2768	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩)
8		vex 3455	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
9		vex 3455	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
10	8, 9	opth 5445	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
11	7, 10	sylbb1 240	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
12		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
13		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
14	12, 13	bi2anan9 650	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ↔ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
15	14	biimpa 482	. . . . . . . . 9 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
16	11, 15	sylan 592	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
17	16	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉))) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
18	17	adantrrr 738	. . . . . 6 ⊢ ((𝜑 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓))) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
19	18	ex 418	. . . . 5 ⊢ (𝜑 → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
20	19	exlimdvv 1967	. . . 4 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
21	20	imp 412	. . 3 ⊢ ((𝜑 ∧ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓))) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
22		simprl 783	. . 3 ⊢ ((𝜑 ∧ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
23		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑈)
24		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
25	14	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ↔ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
26		brab2d.2	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
27	25, 26	anbi12d 644	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
28	27	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
29	23, 24, 28	copsex2dv 5466	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
30	21, 22, 29	bibiad 853	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
31	6, 30	bitrd 282	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103  {copab 5167

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168

This theorem is used by:  tgaaddcpbl2  29353  brprlng  29416  erlcl1  33821  erlcl2  33822  erldi  33823  erlbrd  33824  erler  33826  fracerl  33868