MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpiundi Structured version   Visualization version   GIF version

Theorem xpiundi 5330
Description: Distributive law for Cartesian 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 3237 . . . 4 (∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
2 eliun 4676 . . . . . . . 8 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32anbi1i 733 . . . . . . 7 ((𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
43exbii 1923 . . . . . 6 (∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
5 df-rex 3056 . . . . . 6 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩))
6 df-rex 3056 . . . . . . . 8 (∃𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
76rexbii 3179 . . . . . . 7 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
8 rexcom4 3365 . . . . . . 7 (∃𝑥𝐴𝑦(𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
9 r19.41v 3227 . . . . . . . 8 (∃𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ (∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
109exbii 1923 . . . . . . 7 (∃𝑦𝑥𝐴 (𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩) ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
117, 8, 103bitri 286 . . . . . 6 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑦(∃𝑥𝐴 𝑦𝐵𝑧 = ⟨𝑤, 𝑦⟩))
124, 5, 113bitr4i 292 . . . . 5 (∃𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1312rexbii 3179 . . . 4 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑤𝐶𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
14 elxp2 5289 . . . . 5 (𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
1514rexbii 3179 . . . 4 (∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵) ↔ ∃𝑥𝐴𝑤𝐶𝑦𝐵 𝑧 = ⟨𝑤, 𝑦⟩)
161, 13, 153bitr4i 292 . . 3 (∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩ ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
17 elxp2 5289 . . 3 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ ∃𝑤𝐶𝑦 𝑥𝐴 𝐵𝑧 = ⟨𝑤, 𝑦⟩)
18 eliun 4676 . . 3 (𝑧 𝑥𝐴 (𝐶 × 𝐵) ↔ ∃𝑥𝐴 𝑧 ∈ (𝐶 × 𝐵))
1916, 17, 183bitr4i 292 . 2 (𝑧 ∈ (𝐶 × 𝑥𝐴 𝐵) ↔ 𝑧 𝑥𝐴 (𝐶 × 𝐵))
2019eqriv 2757 1 (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  wrex 3051  cop 4327   ciun 4672   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-iun 4674  df-opab 4865  df-xp 5272
This theorem is referenced by:  xpexgALT  7327  txbasval  21631  txcmplem2  21667  xkoinjcn  21712  cvmlift2lem12  31624
  Copyright terms: Public domain W3C validator