Theorem elinxp 6019

Description: Membership in an intersection with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.)

Assertion

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

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

Proof of Theorem elinxp

Step	Hyp	Ref	Expression
1		relinxp 5802	. . . . 5 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2		elrel 5785	. . . . 5 ⊢ ((Rel (𝑅 ∩ (𝐴 × 𝐵)) ∧ 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
3	1, 2	mpan 702	. . . 4 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
4		eleq1 2857	. . . . . . . . 9 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
5	4	biimpd 232	. . . . . . . 8 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
6		opelinxp 5742	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
7	6	biimpi 219	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8	5, 7	syl6com 38	. . . . . . 7 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
9	8	ancld 559	. . . . . 6 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
10		an12 657	. . . . . 6 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
11	9, 10	imbitrdi 254	. . . . 5 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
12	11	2eximdv 1946	. . . 4 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
13	3, 12	mpd 16	. . 3 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
14		r2ex 3208	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
15	13, 14	sylibr 237	. 2 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
16	6	simplbi2 505	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
17	4	biimprd 251	. . . . 5 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))))
18	16, 17	syl9 78	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)))))
19	18	impd 415	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))))
20	19	rexlimivv 3213	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)))
21	15, 20	impbii 212	1 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095   ∩ cin 3912  ⟨cop 4600   × cxp 5660  Rel wrel 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669

This theorem is referenced by:  elres  6020  elidinxp  6047