Theorem xpiundi 4416

Description: Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)

Assertion

Ref	Expression
xpiundi	⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem xpiundi

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

Step	Hyp	Ref	Expression
1		rexcom 2518	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2		eliun 3682	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	anbi1i 445	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
4	3	exbii 1536	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
5		df-rex 2354	. . . . . 6 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
6		df-rex 2354	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
7	6	rexbii 2373	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
8		rexcom4 2622	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
9		r19.41v 2510	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
10	9	exbii 1536	. . . . . . 7 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
11	7, 8, 10	3bitri 204	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
12	4, 5, 11	3bitr4i 210	. . . . 5 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
13	12	rexbii 2373	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14		elxp2 4381	. . . . 5 ⊢ (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
15	14	rexbii 2373	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
16	1, 13, 15	3bitr4i 210	. . 3 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
17		elxp2 4381	. . 3 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18		eliun 3682	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
19	16, 17, 18	3bitr4i 210	. 2 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
20	19	eqriv 2078	1 ⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)

Syntax hints:   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349  ⟨cop 3401  ∪ ciun 3678   × cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-iun 3680  df-opab 3840  df-xp 4369

This theorem is referenced by:  xpexgALT  5780