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Theorem xpriindi 5144
Description: Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
xpriindi
Distinct variable groups:   ,   ,

Proof of Theorem xpriindi
StepHypRef Expression
1 iineq1 4345 . . . . . . 7
2 0iin 4388 . . . . . . 7
31, 2syl6eq 2514 . . . . . 6
43ineq2d 3699 . . . . 5
5 inv1 3812 . . . . 5
64, 5syl6eq 2514 . . . 4
76xpeq2d 5028 . . 3
8 iineq1 4345 . . . . . 6
9 0iin 4388 . . . . . 6
108, 9syl6eq 2514 . . . . 5
1110ineq2d 3699 . . . 4
12 inv1 3812 . . . 4
1311, 12syl6eq 2514 . . 3
147, 13eqtr4d 2501 . 2
15 xpindi 5141 . . 3
16 xpiindi 5143 . . . 4
1716ineq2d 3699 . . 3
1815, 17syl5eq 2510 . 2
1914, 18pm2.61ine 2770 1
Colors of variables: wff setvar class
Syntax hints:  =wceq 1395  =/=wne 2652   cvv 3109  i^icin 3474   c0 3784  |^|_ciin 4331  X.cxp 5002
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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