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Theorem xpiundi 4555
Description: Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
xpiundi (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem xpiundi
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom 2567 . . . 4 (∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2 eliun 3781 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32anbi1i 451 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
43exbii 1565 . . . . . 6 (∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
5 df-rex 2394 . . . . . 6 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩))
6 df-rex 2394 . . . . . . . 8 (∃𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
76rexbii 2414 . . . . . . 7 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
8 rexcom4 2678 . . . . . . 7 (∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
9 r19.41v 2559 . . . . . . . 8 (∃𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
109exbii 1565 . . . . . . 7 (∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
117, 8, 103bitri 205 . . . . . 6 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
124, 5, 113bitr4i 211 . . . . 5 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1312rexbii 2414 . . . 4 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14 elxp2 4515 . . . . 5 (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1514rexbii 2414 . . . 4 (∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
161, 13, 153bitr4i 211 . . 3 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
17 elxp2 4515 . . 3 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18 eliun 3781 . . 3 (𝑧 𝑥𝐴 (𝐶 × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
1916, 17, 183bitr4i 211 . 2 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ 𝑧 𝑥𝐴 (𝐶 × 𝐵))
2019eqriv 2110 1 (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1312  wex 1449  wcel 1461  wrex 2389  cop 3494   ciun 3777   × cxp 4495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-iun 3779  df-opab 3948  df-xp 4503
This theorem is referenced by:  xpexgALT  5983  txbasval  12272
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