Theorem elinxp 6048

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 5838	. . . . 5 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2		elrel 5822	. . . . 5 ⊢ ((Rel (𝑅 ∩ (𝐴 × 𝐵)) ∧ 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
3	1, 2	mpan 689	. . . 4 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥∃𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
4		eleq1 2832	. . . . . . . . 9 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
5	4	biimpd 229	. . . . . . . 8 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
6		opelinxp 5779	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
7	6	biimpi 216	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
8	5, 7	syl6com 37	. . . . . . 7 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
9	8	ancld 550	. . . . . 6 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
10		an12 644	. . . . . 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 251	. . . . 5 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → (𝐶 = ⟨𝑥, 𝑦⟩ → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))))
12	11	2eximdv 1918	. . . 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 15	. . 3 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅)))
14		r2ex 3202	. . 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 234	. 2 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
16	6	simplbi2 500	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵))))
17	4	biimprd 248	. . . . 5 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))))
18	16, 17	syl9 77	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)))))
19	18	impd 410	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵))))
20	19	rexlimivv 3207	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅) → 𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)))
21	15, 20	impbii 209	1 ⊢ (𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076   ∩ cin 3975  ⟨cop 4654   × cxp 5698  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by:  elres  6049  elidinxp  6073