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

Theorem xpiundir 5061
 Description: Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
xpiundir
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem xpiundir
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3138 . . . . 5
2 df-rex 2823 . . . . . 6
32rexbii 2969 . . . . 5
4 eliun 4336 . . . . . . . 8
54anbi1i 695 . . . . . . 7
6 r19.41v 3019 . . . . . . 7
75, 6bitr4i 252 . . . . . 6
87exbii 1644 . . . . 5
91, 3, 83bitr4ri 278 . . . 4
10 df-rex 2823 . . . 4
11 elxp2 5023 . . . . 5
1211rexbii 2969 . . . 4
139, 10, 123bitr4i 277 . . 3
14 elxp2 5023 . . 3
15 eliun 4336 . . 3
1613, 14, 153bitr4i 277 . 2
1716eqriv 2463 1
 Colors of variables: wff setvar class Syntax hints:   wa 369   wceq 1379  wex 1596   wcel 1767  wrex 2818  cop 4039  ciun 4331   cxp 5003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-iun 4333  df-opab 4512  df-xp 5011 This theorem is referenced by:  iunxpconst  5062  resiun2  5299  txbasval  19975  txtube  20009  txcmplem1  20010  ovoliunlem1  21781
 Copyright terms: Public domain W3C validator