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Theorem xpiundir 4598
 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 2709 . . . . 5
2 df-rex 2422 . . . . . 6
32rexbii 2442 . . . . 5
4 eliun 3817 . . . . . . . 8
54anbi1i 453 . . . . . . 7
6 r19.41v 2587 . . . . . . 7
75, 6bitr4i 186 . . . . . 6
87exbii 1584 . . . . 5
91, 3, 83bitr4ri 212 . . . 4
10 df-rex 2422 . . . 4
11 elxp2 4557 . . . . 5
1211rexbii 2442 . . . 4
139, 10, 123bitr4i 211 . . 3
14 elxp2 4557 . . 3
15 eliun 3817 . . 3
1613, 14, 153bitr4i 211 . 2
1716eqriv 2136 1
 Colors of variables: wff set class Syntax hints:   wa 103   wceq 1331  wex 1468   wcel 1480  wrex 2417  cop 3530  ciun 3813   cxp 4537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iun 3815  df-opab 3990  df-xp 4545 This theorem is referenced by:  iunxpconst  4599  resiun2  4839  txbasval  12450
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