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Theorem ixpiin 7080
 Description: The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)
Assertion
Ref Expression
ixpiin
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem ixpiin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3715 . . . 4
2 vex 2951 . . . . . 6
3 eliin 4090 . . . . . 6
42, 3ax-mp 8 . . . . 5
52elixp 7061 . . . . . 6
65ralbii 2721 . . . . 5
74, 6bitri 241 . . . 4
82elixp 7061 . . . . 5
9 fvex 5734 . . . . . . . . 9
10 eliin 4090 . . . . . . . . 9
119, 10ax-mp 8 . . . . . . . 8
1211ralbii 2721 . . . . . . 7
13 ralcom 2860 . . . . . . 7
1412, 13bitri 241 . . . . . 6
1514anbi2i 676 . . . . 5
168, 15bitri 241 . . . 4
171, 7, 163bitr4g 280 . . 3
1817eqrdv 2433 . 2
1918eqcomd 2440 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725   wne 2598  wral 2697  cvv 2948  c0 3620  ciin 4086   wfn 5441  cfv 5446  cixp 7055 This theorem is referenced by:  ixpint  7081  ptbasfi  17605 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iin 4088  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ixp 7056
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