Theorem xpiindim 4748

Description: Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)

Assertion

Ref	Expression
xpiindim	⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem xpiindim

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

Step	Hyp	Ref	Expression
1		relxp 4720	. . . . . 6 ⊢ Rel (𝐶 × 𝐵)
2	1	rgenw 2525	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)
3		r19.2m 3501	. . . . 5 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
4	2, 3	mpan2 423	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
5		reliin 4733	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵) → Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
6	4, 5	syl 14	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
7		relxp 4720	. . 3 ⊢ Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)
8	6, 7	jctil 310	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
9		eleq1w 2231	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10	9	cbvexv 1911	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
11		r19.28mv 3507	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)))
12	10, 11	sylbir 134	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)))
13	12	bicomd 140	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝑤 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵)))
14		eliin 3878	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
15	14	elv 2734	. . . . . 6 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
16	15	anbi2i 454	. . . . 5 ⊢ ((𝑤 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
17		opelxp 4641	. . . . . 6 ⊢ (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
18	17	ralbii 2476	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
19	13, 16, 18	3bitr4g 222	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝑤 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
20		opelxp 4641	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵))
21		vex 2733	. . . . . 6 ⊢ 𝑤 ∈ V
22		vex 2733	. . . . . 6 ⊢ 𝑧 ∈ V
23	21, 22	opex 4214	. . . . 5 ⊢ ⟨𝑤, 𝑧⟩ ∈ V
24		eliin 3878	. . . . 5 ⊢ (⟨𝑤, 𝑧⟩ ∈ V → (⟨𝑤, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
25	23, 24	ax-mp 5	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵))
26	19, 20, 25	3bitr4g 222	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (⟨𝑤, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ⟨𝑤, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
27	26	eqrelrdv2 4710	. 2 ⊢ (((Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) ∧ ∃𝑦 𝑦 ∈ 𝐴) → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
28	8, 27	mpancom 420	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  Vcvv 2730  ⟨cop 3586  ∩ ciin 3874   × cxp 4609  Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iin 3876  df-opab 4051  df-xp 4617  df-rel 4618

This theorem is referenced by:  xpriindim  4749