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Theorem xpiundir 4566
Description: Distributive law for cross 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 2681 . . . . 5 (∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2 df-rex 2397 . . . . . 6 (∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
32rexbii 2417 . . . . 5 (∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4 eliun 3785 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
54anbi1i 451 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
6 r19.41v 2562 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
75, 6bitr4i 186 . . . . . 6 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
87exbii 1567 . . . . 5 (∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
91, 3, 83bitr4ri 212 . . . 4 (∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
10 df-rex 2397 . . . 4 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11 elxp2 4525 . . . . 5 (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
1211rexbii 2417 . . . 4 (∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
139, 10, 123bitr4i 211 . . 3 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
14 elxp2 4525 . . 3 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ ∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15 eliun 3785 . . 3 (𝑧 𝑥𝐴 (𝐵 × 𝐶) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
1613, 14, 153bitr4i 211 . 2 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ 𝑧 𝑥𝐴 (𝐵 × 𝐶))
1716eqriv 2112 1 ( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wex 1451  wcel 1463  wrex 2392  cop 3498   ciun 3781   × cxp 4505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-iun 3783  df-opab 3958  df-xp 4513
This theorem is referenced by:  iunxpconst  4567  resiun2  4807  txbasval  12331
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