ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpiundir GIF version

Theorem xpiundir 4568
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 2683 . . . . 5 (∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2 df-rex 2399 . . . . . 6 (∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
32rexbii 2419 . . . . 5 (∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4 eliun 3787 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
54anbi1i 453 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
6 r19.41v 2564 . . . . . . 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 1569 . . . . 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 2399 . . . 4 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11 elxp2 4527 . . . . 5 (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
1211rexbii 2419 . . . 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 4527 . . 3 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ ∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15 eliun 3787 . . 3 (𝑧 𝑥𝐴 (𝐵 × 𝐶) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
1613, 14, 153bitr4i 211 . 2 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ 𝑧 𝑥𝐴 (𝐵 × 𝐶))
1716eqriv 2114 1 ( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wex 1453  wcel 1465  wrex 2394  cop 3500   ciun 3783   × cxp 4507
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-iun 3785  df-opab 3960  df-xp 4515
This theorem is referenced by:  iunxpconst  4569  resiun2  4809  txbasval  12363
  Copyright terms: Public domain W3C validator