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Theorem xpiundir 5745
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 3276 . . . . 5 (∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2 df-rex 3061 . . . . . 6 (∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
32rexbii 3084 . . . . 5 (∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4 eliun 4997 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
54anbi1i 622 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
6 r19.41v 3179 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
75, 6bitr4i 277 . . . . . 6 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
87exbii 1843 . . . . 5 (∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
91, 3, 83bitr4ri 303 . . . 4 (∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
10 df-rex 3061 . . . 4 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11 elxp2 5698 . . . . 5 (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
1211rexbii 3084 . . . 4 (∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
139, 10, 123bitr4i 302 . . 3 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
14 elxp2 5698 . . 3 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ ∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15 eliun 4997 . . 3 (𝑧 𝑥𝐴 (𝐵 × 𝐶) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
1613, 14, 153bitr4i 302 . 2 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ 𝑧 𝑥𝐴 (𝐵 × 𝐶))
1716eqriv 2723 1 ( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1534  wex 1774  wcel 2099  wrex 3060  cop 4629   ciun 4993   × cxp 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-iun 4995  df-opab 5208  df-xp 5680
This theorem is referenced by:  iunxpconst  5746  resiun2  6002  txbasval  23598  txtube  23632  txcmplem1  23633  ovoliunlem1  25519
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