Theorem xpiundi 4555

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 2567	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2		eliun 3781	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	anbi1i 451	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
4	3	exbii 1565	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
5		df-rex 2394	. . . . . 6 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
6		df-rex 2394	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
7	6	rexbii 2414	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
8		rexcom4 2678	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
9		r19.41v 2559	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
10	9	exbii 1565	. . . . . . 7 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
11	7, 8, 10	3bitri 205	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑧 = ⟨𝑤, 𝑦⟩))
12	4, 5, 11	3bitr4i 211	. . . . 5 ⊢ (∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
13	12	rexbii 2414	. . . 4 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14		elxp2 4515	. . . . 5 ⊢ (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
15	14	rexbii 2414	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
16	1, 13, 15	3bitr4i 211	. . 3 ⊢ (∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
17		elxp2 4515	. . 3 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑤 ∈ 𝐶 ∃𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18		eliun 3781	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ (𝐶 × 𝐵))
19	16, 17, 18	3bitr4i 211	. 2 ⊢ (𝑧 ∈ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
20	19	eqriv 2110	1 ⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)

Syntax hints:   ∧ wa 103   = wceq 1312  ∃wex 1449   ∈ wcel 1461  ∃wrex 2389  ⟨cop 3494  ∪ ciun 3777   × cxp 4495

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-iun 3779  df-opab 3948  df-xp 4503

This theorem is referenced by:  xpexgALT  5983  txbasval  12272