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Theorem xpiundi 5758
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 3287 . . . 4 (∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2 eliun 4999 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32anbi1i 624 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
43exbii 1844 . . . . . 6 (∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
5 df-rex 3068 . . . . . 6 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩))
6 df-rex 3068 . . . . . . . 8 (∃𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
76rexbii 3091 . . . . . . 7 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
8 rexcom4 3285 . . . . . . 7 (∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
9 r19.41v 3186 . . . . . . . 8 (∃𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
109exbii 1844 . . . . . . 7 (∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
117, 8, 103bitri 297 . . . . . 6 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
124, 5, 113bitr4i 303 . . . . 5 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1312rexbii 3091 . . . 4 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14 elxp2 5712 . . . . 5 (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1514rexbii 3091 . . . 4 (∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
161, 13, 153bitr4i 303 . . 3 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
17 elxp2 5712 . . 3 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18 eliun 4999 . . 3 (𝑧 𝑥𝐴 (𝐶 × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
1916, 17, 183bitr4i 303 . 2 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ 𝑧 𝑥𝐴 (𝐶 × 𝐵))
2019eqriv 2731 1 (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wex 1775  wcel 2105  wrex 3067  cop 4636   ciun 4995   × cxp 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-iun 4997  df-opab 5210  df-xp 5694
This theorem is referenced by:  xpexgALT  8004  txbasval  23629  txcmplem2  23665  xkoinjcn  23710  cvmlift2lem12  35298
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