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Theorem xpiundi 5322
Description: Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
xpiundi (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (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.
StepHypRef Expression
1 rexcom 3229 . . . 4 (∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2 eliun 4668 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32anbi1i 733 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
43exbii 1915 . . . . . 6 (∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
5 df-rex 3048 . . . . . 6 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩))
6 df-rex 3048 . . . . . . . 8 (∃𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
76rexbii 3171 . . . . . . 7 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
8 rexcom4 3357 . . . . . . 7 (∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
9 r19.41v 3219 . . . . . . . 8 (∃𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
109exbii 1915 . . . . . . 7 (∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
117, 8, 103bitri 286 . . . . . 6 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
124, 5, 113bitr4i 292 . . . . 5 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1312rexbii 3171 . . . 4 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14 elxp2 5281 . . . . 5 (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1514rexbii 3171 . . . 4 (∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
161, 13, 153bitr4i 292 . . 3 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
17 elxp2 5281 . . 3 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18 eliun 4668 . . 3 (𝑧 𝑥𝐴 (𝐶 × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
1916, 17, 183bitr4i 292 . 2 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ 𝑧 𝑥𝐴 (𝐶 × 𝐵))
2019eqriv 2749 1 (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1624  wex 1845  wcel 2131  wrex 3043  cop 4319   ciun 4664   × cxp 5256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-iun 4666  df-opab 4857  df-xp 5264
This theorem is referenced by:  xpexgALT  7318  txbasval  21603  txcmplem2  21639  xkoinjcn  21684  cvmlift2lem12  31595
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