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Theorem xpiindi 5143
Description: Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
xpiindi
Distinct variable groups:   ,   ,

Proof of Theorem xpiindi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5115 . . . . . 6
21rgenw 2818 . . . . 5
3 r19.2z 3918 . . . . 5
42, 3mpan2 671 . . . 4
5 reliin 5129 . . . 4
64, 5syl 16 . . 3
7 relxp 5115 . . 3
86, 7jctil 537 . 2
9 r19.28zv 3924 . . . . . 6
109bicomd 201 . . . . 5
11 vex 3112 . . . . . . 7
12 eliin 4336 . . . . . . 7
1311, 12ax-mp 5 . . . . . 6
1413anbi2i 694 . . . . 5
15 opelxp 5034 . . . . . 6
1615ralbii 2888 . . . . 5
1710, 14, 163bitr4g 288 . . . 4
18 opelxp 5034 . . . 4
19 opex 4716 . . . . 5
20 eliin 4336 . . . . 5
2119, 20ax-mp 5 . . . 4
2217, 18, 213bitr4g 288 . . 3
2322eqrelrdv2 5107 . 2
248, 23mpancom 669 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808   cvv 3109   c0 3784  <.cop 4035  |^|_ciin 4331  X.cxp 5002  Relwrel 5009
This theorem is referenced by:  xpriindi  5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-iin 4333  df-opab 4511  df-xp 5010  df-rel 5011
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