Theorem iinxp 50050

Description: Indexed intersection of Cartesian products is the Cartesian product of indexed intersections. See also inxp 5810 and intxpd 50051. (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 5669	. . . . 5 ⊢ Rel (𝐵 × 𝐶)
2	1	rgenw 3080	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶)
3		r19.2z 4454	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶)) → ∃𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶))
4	2, 3	mpan2 704	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 Rel (𝐵 × 𝐶))
5		reliin 5796	. . 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 5669	. 2 ⊢ Rel (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶)
8		eliin 4955	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
9	8	elv 3455	. . . . 5 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
10		eliin 4955	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
11	10	elv 3455	. . . . 5 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
12	9, 11	anbi12i 640	. . . 4 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐶))
13		opelxp 5687	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
14		opex 5432	. . . . . 6 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
15		eliin 4955	. . . . . 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 5687	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
18	17	ralbii 3108	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
19		r19.26 3122	. . . . 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 5766	. 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 598	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450  ∅c0 4277  ⟨cop 4834  ∩ ciin 4951   × cxp 5649  Rel wrel 5656

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-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-iin 4953  df-opab 5167  df-xp 5657  df-rel 5658

This theorem is used by:  intxpd  50051  iinfssclem1  50276