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Theorem xpiundir 5759
Description: Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
xpiundir ( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem xpiundir
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3285 . . . . 5 (∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2 df-rex 3068 . . . . . 6 (∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
32rexbii 3091 . . . . 5 (∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4 eliun 4999 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
54anbi1i 624 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
6 r19.41v 3186 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
75, 6bitr4i 278 . . . . . 6 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
87exbii 1844 . . . . 5 (∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
91, 3, 83bitr4ri 304 . . . 4 (∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
10 df-rex 3068 . . . 4 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11 elxp2 5712 . . . . 5 (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
1211rexbii 3091 . . . 4 (∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
139, 10, 123bitr4i 303 . . 3 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
14 elxp2 5712 . . 3 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ ∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15 eliun 4999 . . 3 (𝑧 𝑥𝐴 (𝐵 × 𝐶) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
1613, 14, 153bitr4i 303 . 2 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ 𝑧 𝑥𝐴 (𝐵 × 𝐶))
1716eqriv 2731 1 ( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)
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:  iunxpconst  5760  resiun2  6020  txbasval  23629  txtube  23663  txcmplem1  23664  ovoliunlem1  25550
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