Theorem xpiundi 5096

Description: Distributive law for Cartesian 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 3080	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2		eliun 4460	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	anbi1i 727	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
4	3	exbii 1764	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
5		df-rex 2902	. . . . . 6 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
6		df-rex 2902	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
7	6	rexbii 3023	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
8		rexcom4 3198	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
9		r19.41v 3070	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
10	9	exbii 1764	. . . . . . 7 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
11	7, 8, 10	3bitri 285	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
12	4, 5, 11	3bitr4i 291	. . . . 5 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
13	12	rexbii 3023	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14		elxp2 5056	. . . . 5 ⊢ (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
15	14	rexbii 3023	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
16	1, 13, 15	3bitr4i 291	. . 3 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
17		elxp2 5056	. . 3 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18		eliun 4460	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
19	16, 17, 18	3bitr4i 291	. 2 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
20	19	eqriv 2607	1 ⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131  ∪ ciun 4455   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-iun 4457  df-opab 4644  df-xp 5044

This theorem is referenced by:  xpexgALT  7052  txbasval  21219  txcmplem2  21255  xkoinjcn  21300  cvmlift2lem12  30550