Theorem xpiundi 4786

Description: Distributive law for cross 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 2696	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2		eliun 3975	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	anbi1i 458	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
4	3	exbii 1653	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
5		df-rex 2515	. . . . . 6 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
6		df-rex 2515	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
7	6	rexbii 2538	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
8		rexcom4 2825	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
9		r19.41v 2688	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
10	9	exbii 1653	. . . . . . 7 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
11	7, 8, 10	3bitri 206	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
12	4, 5, 11	3bitr4i 212	. . . . 5 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
13	12	rexbii 2538	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14		elxp2 4745	. . . . 5 ⊢ (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
15	14	rexbii 2538	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
16	1, 13, 15	3bitr4i 212	. . 3 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
17		elxp2 4745	. . 3 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18		eliun 3975	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
19	16, 17, 18	3bitr4i 212	. 2 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
20	19	eqriv 2227	1 ⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2201  ∃wrex 2510  ⟨cop 3673  ∪ ciun 3971   × cxp 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-iun 3973  df-opab 4152  df-xp 4733

This theorem is referenced by:  xpexgALT  6300  txbasval  15020