Theorem xpiundir 5669

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

Assertion

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

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem xpiundir

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

Step	Hyp	Ref	Expression
1		rexcom4 3268	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2		df-rex 3072	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
3	2	rexbii 3094	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4		eliun 4935	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5	4	anbi1i 625	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
6		r19.41v 3182	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
7	5, 6	bitr4i 278	. . . . . 6 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
8	7	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
9	1, 3, 8	3bitr4ri 304	. . . 4 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
10		df-rex 3072	. . . 4 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11		elxp2 5624	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
12	11	rexbii 3094	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
13	9, 10, 12	3bitr4i 303	. . 3 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶))
14		elxp2 5624	. . 3 ⊢ (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) ↔ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15		eliun 4935	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶))
16	13, 14, 15	3bitr4i 303	. 2 ⊢ (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
17	16	eqriv 2733	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶)

Syntax hints:   ∧ wa 397   = wceq 1539  ∃wex 1779   ∈ wcel 2104  ∃wrex 3071  ⟨cop 4571  ∪ ciun 4931   × cxp 5598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-iun 4933  df-opab 5144  df-xp 5606

This theorem is referenced by:  iunxpconst  5670  resiun2  5924  txbasval  22806  txtube  22840  txcmplem1  22841  ovoliunlem1  24715