Theorem xpiundir 4718

Description: Distributive law for cross 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 2783	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2		df-rex 2478	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
3	2	rexbii 2501	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4		eliun 3916	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5	4	anbi1i 458	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
6		r19.41v 2650	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
7	5, 6	bitr4i 187	. . . . . 6 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
8	7	exbii 1616	. . . . 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 213	. . . 4 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
10		df-rex 2478	. . . 4 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11		elxp2 4677	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
12	11	rexbii 2501	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
13	9, 10, 12	3bitr4i 212	. . 3 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶))
14		elxp2 4677	. . 3 ⊢ (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) ↔ ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∃𝑤 ∈ 𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15		eliun 3916	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐵 × 𝐶))
16	13, 14, 15	3bitr4i 212	. 2 ⊢ (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
17	16	eqriv 2190	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶)

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∃wrex 2473  ⟨cop 3621  ∪ ciun 3912   × cxp 4657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-iun 3914  df-opab 4091  df-xp 4665

This theorem is referenced by:  iunxpconst  4719  resiun2  4962  txbasval  14435