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Theorem xpiundi 5134
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 3091 . . . 4 (∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2 eliun 4490 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32anbi1i 730 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
43exbii 1771 . . . . . 6 (∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
5 df-rex 2913 . . . . . 6 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩))
6 df-rex 2913 . . . . . . . 8 (∃𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
76rexbii 3034 . . . . . . 7 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
8 rexcom4 3211 . . . . . . 7 (∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
9 r19.41v 3081 . . . . . . . 8 (∃𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
109exbii 1771 . . . . . . 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 3034 . . . 4 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14 elxp2 5092 . . . . 5 (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1514rexbii 3034 . . . 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 5092 . . 3 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18 eliun 4490 . . 3 (𝑧 𝑥𝐴 (𝐶 × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
1916, 17, 183bitr4i 292 . 2 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ 𝑧 𝑥𝐴 (𝐶 × 𝐵))
2019eqriv 2618 1 (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  cop 4154   ciun 4485   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-iun 4487  df-opab 4674  df-xp 5080
This theorem is referenced by:  xpexgALT  7106  txbasval  21319  txcmplem2  21355  xkoinjcn  21400  cvmlift2lem12  31001
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