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Theorem xpiundi 5063
 Description: Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
xpiundi
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem xpiundi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom 3019 . . . 4
2 eliun 4337 . . . . . . . 8
32anbi1i 695 . . . . . . 7
43exbii 1668 . . . . . 6
5 df-rex 2813 . . . . . 6
6 df-rex 2813 . . . . . . . 8
76rexbii 2959 . . . . . . 7
8 rexcom4 3129 . . . . . . 7
9 r19.41v 3009 . . . . . . . 8
109exbii 1668 . . . . . . 7
117, 8, 103bitri 271 . . . . . 6
124, 5, 113bitr4i 277 . . . . 5
1312rexbii 2959 . . . 4
14 elxp2 5026 . . . . 5
1514rexbii 2959 . . . 4
161, 13, 153bitr4i 277 . . 3
17 elxp2 5026 . . 3
18 eliun 4337 . . 3
1916, 17, 183bitr4i 277 . 2
2019eqriv 2453 1
 Colors of variables: wff setvar class Syntax hints:   wa 369   wceq 1395  wex 1613   wcel 1819  wrex 2808  cop 4038  ciun 4332   cxp 5006 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  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 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-iun 4334  df-opab 4516  df-xp 5014 This theorem is referenced by:  xpexgALT  6792  txbasval  20233  txcmplem2  20269  xkoinjcn  20314  cvmlift2lem12  28956
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