Theorem xpiindi 5774

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 5632	. . . . . 6 ⊢ Rel (𝐶 × 𝐵)
2	1	rgenw 3051	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)
3		r19.2z 4442	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
4	2, 3	mpan2 691	. . . 4 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
5		reliin 5756	. . . 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 5632	. . 3 ⊢ Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)
8	6, 7	jctil 519	. 2 ⊢ (𝐴 ≠ ∅ → (Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
9		r19.28zv 4448	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)))
10	9	bicomd 223	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵)))
11		eliin 4944	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
12	11	elv 3441	. . . . . 6 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
13	12	anbi2i 623	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
14		opelxp 5650	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
15	14	ralbii 3078	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
16	10, 13, 15	3bitr4g 314	. . . 4 ⊢ (𝐴 ≠ ∅ → ((𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ (𝐶 × 𝐵)))
17		opelxp 5650	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵))
18		opex 5402	. . . . 5 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
19		eliin 4944	. . . . 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 314	. . 3 ⊢ (𝐴 ≠ ∅ → (⟨𝑦, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
22	21	eqrelrdv2 5734	. 2 ⊢ (((Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) ∧ 𝐴 ≠ ∅) → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
23	8, 22	mpancom 688	1 ⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  Vcvv 3436  ∅c0 4280  ⟨cop 4579  ∩ ciin 4940   × cxp 5612  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-iin 4942  df-opab 5152  df-xp 5620  df-rel 5621

This theorem is referenced by:  xpriindi  5775