Theorem xpiundi 5586

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 3308	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2		eliun 4885	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	anbi1i 626	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
4	3	exbii 1849	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
5		df-rex 3112	. . . . . 6 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
6		df-rex 3112	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
7	6	rexbii 3210	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
8		rexcom4 3212	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
9		r19.41v 3300	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
10	9	exbii 1849	. . . . . . 7 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
11	7, 8, 10	3bitri 300	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
12	4, 5, 11	3bitr4i 306	. . . . 5 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
13	12	rexbii 3210	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14		elxp2 5543	. . . . 5 ⊢ (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
15	14	rexbii 3210	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
16	1, 13, 15	3bitr4i 306	. . 3 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
17		elxp2 5543	. . 3 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18		eliun 4885	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
19	16, 17, 18	3bitr4i 306	. 2 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
20	19	eqriv 2795	1 ⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)

Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  ⟨cop 4531  ∪ ciun 4881   × cxp 5517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4883  df-opab 5093  df-xp 5525

This theorem is referenced by:  xpexgALT  7664  txbasval  22211  txcmplem2  22247  xkoinjcn  22292  cvmlift2lem12  32674