Theorem iinxp 48718

Description: Indexed intersection of Cartesian products is the Cartesian product of indexed intersections. See also inxp 5822 and intxp 48719. (Contributed by Zhi Wang, 30-Oct-2025.)

Assertion

Ref	Expression
iinxp	⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iinxp

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5683	. . . . 5 ⊢ Rel (𝐵 × 𝐶)
2	1	rgenw 3054	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶)
3		r19.2z 4475	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶)) → ∃𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶))
4	2, 3	mpan2 691	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶))
5		reliin 5807	. . 3 ⊢ (∃𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶) → Rel ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
6	4, 5	syl 17	. 2 ⊢ (𝐴 ≠ ∅ → Rel ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
7		relxp 5683	. 2 ⊢ Rel (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶)
8		eliin 4976	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
9	8	elv 3468	. . . . 5 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
10		eliin 4976	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
11	10	elv 3468	. . . . 5 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
12	9, 11	anbi12i 628	. . . 4 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
13		opelxp 5701	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
14		opex 5449	. . . . . 6 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
15		eliin 4976	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ V → (⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)))
16	14, 15	ax-mp 5	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶))
17		opelxp 5701	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
18	17	ralbii 3081	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
19		r19.26 3098	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
20	16, 18, 19	3bitri 297	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
21	12, 13, 20	3bitr4ri 304	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))
22	21	eqrelriv 5779	. 2 ⊢ ((Rel ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ∧ Rel (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶)) → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))
23	6, 7, 22	sylancl 586	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  Vcvv 3463  ∅c0 4313  ⟨cop 4612  ∩ ciin 4972   × cxp 5663  Rel wrel 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-iin 4974  df-opab 5186  df-xp 5671  df-rel 5672

This theorem is referenced by:  intxp  48719  iinfssclem1  48927