Theorem iinxp 49493

Description: Indexed intersection of Cartesian products is the Cartesian product of indexed intersections. See also inxp 5819 and intxp 49494. (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 5680	. . . . 5 ⊢ Rel (𝐵 × 𝐶)
2	1	rgenw 3089	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶)
3		r19.2z 4465	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶)) → ∃𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶))
4	2, 3	mpan2 703	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶))
5		reliin 5805	. . 3 ⊢ (∃𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶) → Rel ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
6	4, 5	syl 18	. 2 ⊢ (𝐴 ≠ ∅ → Rel ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
7		relxp 5680	. 2 ⊢ Rel (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶)
8		eliin 4965	. . . . . 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 4965	. . . . . 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 639	. . . 4 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
13		opelxp 5698	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
14		opex 5446	. . . . . 6 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
15		eliin 4965	. . . . . 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 5698	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
18	17	ralbii 3117	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
19		r19.26 3131	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
20	16, 18, 19	3bitri 300	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
21	12, 13, 20	3bitr4ri 307	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))
22	21	eqrelriv 5776	. 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 597	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463  ∅c0 4294  ⟨cop 4600  ∩ ciin 4961   × 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-ne 2965  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-iin 4963  df-opab 5178  df-xp 5668  df-rel 5669

This theorem is referenced by:  intxp  49494  iinfssclem1  49716