Theorem xpiindi 5842

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 5700	. . . . . 6 ⊢ Rel (𝐶 × 𝐵)
2	1	rgenw 3062	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)
3		r19.2z 4498	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
4	2, 3	mpan2 689	. . . 4 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
5		reliin 5823	. . . 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 5700	. . 3 ⊢ Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)
8	6, 7	jctil 518	. 2 ⊢ (𝐴 ≠ ∅ → (Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
9		r19.28zv 4504	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)))
10	9	bicomd 222	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵)))
11		eliin 5005	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
12	11	elv 3479	. . . . . 6 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
13	12	anbi2i 621	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
14		opelxp 5718	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
15	14	ralbii 3090	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
16	10, 13, 15	3bitr4g 313	. . . 4 ⊢ (𝐴 ≠ ∅ → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
17		opelxp 5718	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵))
18		opex 5470	. . . . 5 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
19		eliin 5005	. . . . 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 313	. . 3 ⊢ (𝐴 ≠ ∅ → (⟨𝑦, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
22	21	eqrelrdv2 5801	. 2 ⊢ (((Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) ∧ 𝐴 ≠ ∅) → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
23	8, 22	mpancom 686	1 ⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3473  ∅c0 4326  ⟨cop 4638  ∩ ciin 5001   × cxp 5680  Rel wrel 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-iin 5003  df-opab 5215  df-xp 5688  df-rel 5689

This theorem is referenced by:  xpriindi  5843