Theorem xpiundir 5737

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 3277	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2		df-rex 3063	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
3	2	rexbii 3086	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4		eliun 4991	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5	4	anbi1i 623	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
6		r19.41v 3180	. . . . . . 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 1842	. . . . 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 3063	. . . 4 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11		elxp2 5690	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
12	11	rexbii 3086	. . . 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 5690	. . 3 ⊢ (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) ↔ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15		eliun 4991	. . 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 2721	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃wrex 3062  ⟨cop 4626  ∪ ciun 4987   × cxp 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-iun 4989  df-opab 5201  df-xp 5672

This theorem is referenced by:  iunxpconst  5738  resiun2  5992  txbasval  23431  txtube  23465  txcmplem1  23466  ovoliunlem1  25352