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Theorem xpiundir 4976
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 3072 . . . . 5
2 df-rex 2798 . . . . . 6
32rexbii 2813 . . . . 5
4 eliun 4257 . . . . . . . 8
54anbi1i 695 . . . . . . 7
6 r19.41v 2953 . . . . . . 7
75, 6bitr4i 252 . . . . . 6
87exbii 1635 . . . . 5
91, 3, 83bitr4ri 278 . . . 4
10 df-rex 2798 . . . 4
11 elxp2 4940 . . . . 5
1211rexbii 2813 . . . 4
139, 10, 123bitr4i 277 . . 3
14 elxp2 4940 . . 3
15 eliun 4257 . . 3
1613, 14, 153bitr4i 277 . 2
1716eqriv 2446 1
Colors of variables: wff setvar class
Syntax hints:  /\wa 369  =wceq 1370  E.wex 1587  e.wcel 1757  E.wrex 2793  <.cop 3965  U_ciun 4253  X.cxp 4920
This theorem is referenced by:  iunxpconst  4977  resiun2  5212  txbasval  19279  txtube  19313  txcmplem1  19314  ovoliunlem1  21085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4495  ax-nul 4503  ax-pr 4613
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-v 3054  df-dif 3413  df-un 3415  df-in 3417  df-ss 3424  df-nul 3720  df-if 3874  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4255  df-opab 4433  df-xp 4928
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