Theorem xpiundir 5625

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 3251	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2		df-rex 3146	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
3	2	rexbii 3249	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4		eliun 4925	. . . . . . . 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 3349	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
7	5, 6	bitr4i 280	. . . . . 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 306	. . . 4 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
10		df-rex 3146	. . . 4 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11		elxp2 5581	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
12	11	rexbii 3249	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
13	9, 10, 12	3bitr4i 305	. . 3 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶))
14		elxp2 5581	. . 3 ⊢ (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) ↔ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15		eliun 4925	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶))
16	13, 14, 15	3bitr4i 305	. 2 ⊢ (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
17	16	eqriv 2820	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶)

Syntax hints:   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃wrex 3141  ⟨cop 4575  ∪ ciun 4921   × cxp 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-iun 4923  df-opab 5131  df-xp 5563

This theorem is referenced by:  iunxpconst  5626  resiun2  5876  txbasval  22216  txtube  22250  txcmplem1  22251  ovoliunlem1  24105