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Theorem xpiundir 4399
 Description: Distributive law for cross 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 2577 . . . . 5
2 df-rex 2312 . . . . . 6
32rexbii 2331 . . . . 5
4 eliun 3661 . . . . . . . 8
54anbi1i 431 . . . . . . 7
6 r19.41v 2466 . . . . . . 7
75, 6bitr4i 176 . . . . . 6
87exbii 1496 . . . . 5
91, 3, 83bitr4ri 202 . . . 4
10 df-rex 2312 . . . 4
11 elxp2 4363 . . . . 5
1211rexbii 2331 . . . 4
139, 10, 123bitr4i 201 . . 3
14 elxp2 4363 . . 3
15 eliun 3661 . . 3
1613, 14, 153bitr4i 201 . 2
1716eqriv 2037 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243  wex 1381   wcel 1393  wrex 2307  cop 3378  ciun 3657   cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-opab 3819  df-xp 4351 This theorem is referenced by:  iunxpconst  4400  resiun2  4631
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