Theorem bj-elinxp 38177

Description: Membership in an intersection with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.) Use the 𝑥𝑅𝑦 idiom and give a shorter proof not depending on df-rel 5665. (Revised by BJ, 16-Oct-2026.) Note to remove when moving to main section: a shortening of the exact statement elinxp 6026 is possible along the same line.

Assertion

Ref	Expression
bj-elinxp	⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑅,𝑦

Proof of Theorem bj-elinxp

Step	Hyp	Ref	Expression
1		elin 4096	. 2 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ (𝐶 ∈ 𝑅 ∧ 𝐶 ∈ (𝐴 × 𝐵)))
2		elxp2 5682	. . 3 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩)
3	2	anbi2i 635	. 2 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐶 ∈ (𝐴 × 𝐵)) ↔ (𝐶 ∈ 𝑅 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩))
4		r19.42v 3198	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ (𝐶 ∈ 𝑅 ∧ ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩))
5	4	rexbii 3113	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥 ∈ 𝐴 (𝐶 ∈ 𝑅 ∧ ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩))
6		eleq1 2852	. . . . . . . 8 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
7		df-br 5109	. . . . . . . 8 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
8	6, 7	bitr4di 292	. . . . . . 7 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ 𝑅 ↔ 𝑥𝑅𝑦))
9	8	pm5.32ri 586	. . . . . 6 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ (𝑥𝑅𝑦 ∧ 𝐶 = ⟨𝑥, 𝑦⟩))
10	9	biancomi 468	. . . . 5 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
11	10	rexbii 3113	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
12	11	rexbii 3113	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
13		r19.42v 3198	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝐶 ∈ 𝑅 ∧ ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩) ↔ (𝐶 ∈ 𝑅 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩))
14	5, 12, 13	3bitr3ri 305	. 2 ⊢ ((𝐶 ∈ 𝑅 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
15	1, 3, 14	3bitri 300	1 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ∩ cin 4092  ⟨cop 4840   class class class wbr 5108   × cxp 5656

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664

This theorem is used by:  bj-elres  38184