Theorem xpiindi 5670

Description: Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem xpiindi

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

Step	Hyp	Ref	Expression
1		relxp 5537	. . . . . 6 ⊢ Rel (𝐶 × 𝐵)
2	1	rgenw 3118	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)
3		r19.2z 4398	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
4	2, 3	mpan2 690	. . . 4 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
5		reliin 5654	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵) → Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ≠ ∅ → Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
7		relxp 5537	. . 3 ⊢ Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)
8	6, 7	jctil 523	. 2 ⊢ (𝐴 ≠ ∅ → (Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
9		r19.28zv 4404	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)))
10	9	bicomd 226	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵)))
11		eliin 4886	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
12	11	elv 3446	. . . . . 6 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
13	12	anbi2i 625	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
14		opelxp 5555	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
15	14	ralbii 3133	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
16	10, 13, 15	3bitr4g 317	. . . 4 ⊢ (𝐴 ≠ ∅ → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
17		opelxp 5555	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵))
18		opex 5321	. . . . 5 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
19		eliin 4886	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ V → (⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
20	18, 19	ax-mp 5	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵))
21	16, 17, 20	3bitr4g 317	. . 3 ⊢ (𝐴 ≠ ∅ → (⟨𝑦, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
22	21	eqrelrdv2 5632	. 2 ⊢ (((Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) ∧ 𝐴 ≠ ∅) → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
23	8, 22	mpancom 687	1 ⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441  ∅c0 4243  ⟨cop 4531  ∩ ciin 4882   × cxp 5517  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-iin 4884  df-opab 5093  df-xp 5525  df-rel 5526

This theorem is referenced by:  xpriindi  5671