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Theorem xpiundi 4341
 Description: Distributive law for cross 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 2468 . . . 4
2 eliun 3652 . . . . . . . 8
32anbi1i 431 . . . . . . 7
43exbii 1493 . . . . . 6
5 df-rex 2306 . . . . . 6
6 df-rex 2306 . . . . . . . 8
76rexbii 2325 . . . . . . 7
8 rexcom4 2571 . . . . . . 7
9 r19.41v 2460 . . . . . . . 8
109exbii 1493 . . . . . . 7
117, 8, 103bitri 195 . . . . . 6
124, 5, 113bitr4i 201 . . . . 5
1312rexbii 2325 . . . 4
14 elxp2 4306 . . . . 5
1514rexbii 2325 . . . 4
161, 13, 153bitr4i 201 . . 3
17 elxp2 4306 . . 3
18 eliun 3652 . . 3
1916, 17, 183bitr4i 201 . 2
2019eqriv 2034 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1242  wex 1378   wcel 1390  wrex 2301  cop 3370  ciun 3648   cxp 4286 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-opab 3810  df-xp 4294 This theorem is referenced by:  xpexgALT  5702
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