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Theorem xpiundir 4426
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 2594 . . . . 5 (∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
2 df-rex 2329 . . . . . 6 (∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
32rexbii 2348 . . . . 5 (∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
4 eliun 3688 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
54anbi1i 439 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
6 r19.41v 2483 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ (∃𝑥𝐴 𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
75, 6bitr4i 180 . . . . . 6 ((𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
87exbii 1512 . . . . 5 (∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
91, 3, 83bitr4ri 206 . . . 4 (∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩) ↔ ∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
10 df-rex 2329 . . . 4 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵 ∧ ∃𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩))
11 elxp2 4390 . . . . 5 (𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
1211rexbii 2348 . . . 4 (∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐴𝑦𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
139, 10, 123bitr4i 205 . . 3 (∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩ ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
14 elxp2 4390 . . 3 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ ∃𝑦 𝑥𝐴 𝐵𝑤𝐶 𝑧 = ⟨𝑦, 𝑤⟩)
15 eliun 3688 . . 3 (𝑧 𝑥𝐴 (𝐵 × 𝐶) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐵 × 𝐶))
1613, 14, 153bitr4i 205 . 2 (𝑧 ∈ ( 𝑥𝐴 𝐵 × 𝐶) ↔ 𝑧 𝑥𝐴 (𝐵 × 𝐶))
1716eqriv 2053 1 ( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  wrex 2324  cop 3405   ciun 3684   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-iun 3686  df-opab 3846  df-xp 4378
This theorem is referenced by:  iunxpconst  4427  resiun2  4658
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