Theorem bj-elinxp 38056

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 5658. (Revised by BJ, 16-Oct-2026.) Note to remove when moving to main section: a shortening of the exact statement elinxp 6010 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 4092	. 2 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ (𝐶 ∈ 𝑅 ∧ 𝐶 ∈ (𝐴 × 𝐵)))
2		elxp2 5675	. . 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 3194	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ (𝐶 ∈ 𝑅 ∧ ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩))
5	4	rexbii 3109	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥 ∈ 𝐴 (𝐶 ∈ 𝑅 ∧ ∃𝑦 ∈ 𝐵 𝐶 = ⟨𝑥, 𝑦⟩))
6		eleq1 2848	. . . . . . . 8 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
7		df-br 5103	. . . . . . . 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 3109	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
12	11	rexbii 3109	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 ∈ 𝑅 ∧ 𝐶 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
13		r19.42v 3194	. . 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 1569   ∈ wcel 2144  ∃wrex 3086   ∩ cin 4088  ⟨cop 4834   class class class wbr 5102   × cxp 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-un 4150  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657

This theorem is used by:  bj-elres  38057