MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpiundir Unicode version

Theorem xpiundir 5060
Description: Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
xpiundir
Distinct variable group:   ,

Proof of Theorem xpiundir
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3129 . . . . 5
2 df-rex 2813 . . . . . 6
32rexbii 2959 . . . . 5
4 eliun 4335 . . . . . . . 8
54anbi1i 695 . . . . . . 7
6 r19.41v 3009 . . . . . . 7
75, 6bitr4i 252 . . . . . 6
87exbii 1667 . . . . 5
91, 3, 83bitr4ri 278 . . . 4
10 df-rex 2813 . . . 4
11 elxp2 5022 . . . . 5
1211rexbii 2959 . . . 4
139, 10, 123bitr4i 277 . . 3
14 elxp2 5022 . . 3
15 eliun 4335 . . 3
1613, 14, 153bitr4i 277 . 2
1716eqriv 2453 1
Colors of variables: wff setvar class
Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  <.cop 4035  U_ciun 4330  X.cxp 5002
This theorem is referenced by:  iunxpconst  5061  resiun2  5298  txbasval  20107  txtube  20141  txcmplem1  20142  ovoliunlem1  21913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-iun 4332  df-opab 4511  df-xp 5010
  Copyright terms: Public domain W3C validator