Theorem brab2d 32580

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 5087	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		brab2d.1	. . . . 5 ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)})
3	2	eleq2d 2817	. . . 4 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)}))
4	1, 3	bitrid 283	. . 3 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)}))
5		elopab 5462	. . 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 287	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓))))
7		eqcom 2738	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩)
8		vex 3440	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
9		vex 3440	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
10	8, 9	opth 5411	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
11	7, 10	sylbb1 237	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
12		eleq1 2819	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
13		eleq1 2819	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
14	12, 13	bi2anan9 638	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ↔ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
15	14	biimpa 476	. . . . . . . . 9 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
16	11, 15	sylan 580	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
17	16	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉))) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
18	17	adantrrr 725	. . . . . 6 ⊢ ((𝜑 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓))) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
19	18	ex 412	. . . . 5 ⊢ (𝜑 → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
20	19	exlimdvv 1935	. . . 4 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))
21	20	imp 406	. . 3 ⊢ ((𝜑 ∧ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓))) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
22		simprl 770	. . 3 ⊢ ((𝜑 ∧ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
23		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑈)
24		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
25	14	adantl 481	. . . . . 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 632	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
28	27	adantlr 715	. . . 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 5429	. . 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 839	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)) ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
31	6, 30	bitrd 279	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ⟨cop 4577   class class class wbr 5086  {copab 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149

This theorem is referenced by:  erlcl1  33219  erlcl2  33220  erldi  33221  erlbrd  33222  erler  33224  fracerl  33264