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Theorem dfiin2g 3681
Description: Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
Assertion
Ref Expression
dfiin2g  C  |^|_  |^| {  |  }
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()    C(,)

Proof of Theorem dfiin2g
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ral 2305 . . . 4
2 df-ral 2305 . . . . . 6  C  C
3 eleq2 2098 . . . . . . . . . . . . 13
43biimprcd 149 . . . . . . . . . . . 12
54alrimiv 1751 . . . . . . . . . . 11
6 eqid 2037 . . . . . . . . . . . 12
7 eqeq1 2043 . . . . . . . . . . . . . 14
87, 3imbi12d 223 . . . . . . . . . . . . 13
98spcgv 2634 . . . . . . . . . . . 12  C
106, 9mpii 39 . . . . . . . . . . 11  C
115, 10impbid2 131 . . . . . . . . . 10  C
1211imim2i 12 . . . . . . . . 9  C
1312pm5.74d 171 . . . . . . . 8  C
1413alimi 1341 . . . . . . 7  C
15 albi 1354 . . . . . . 7
1614, 15syl 14 . . . . . 6  C
172, 16sylbi 114 . . . . 5  C
18 df-ral 2305 . . . . . . . 8
1918albii 1356 . . . . . . 7
20 alcom 1364 . . . . . . 7
2119, 20bitr4i 176 . . . . . 6
22 r19.23v 2419 . . . . . . . 8
23 vex 2554 . . . . . . . . . 10 
_V
24 eqeq1 2043 . . . . . . . . . . 11
2524rexbidv 2321 . . . . . . . . . 10
2623, 25elab 2681 . . . . . . . . 9  {  |  }
2726imbi1i 227 . . . . . . . 8  {  |  }
2822, 27bitr4i 176 . . . . . . 7  {  |  }
2928albii 1356 . . . . . 6 
{  |  }
30 19.21v 1750 . . . . . . 7
3130albii 1356 . . . . . 6
3221, 29, 313bitr3ri 200 . . . . 5  {  |  }
3317, 32syl6bb 185 . . . 4  C  {  |  }
341, 33syl5bb 181 . . 3  C  {  |  }
3534abbidv 2152 . 2  C  {  |  }  {  |  {  |  }  }
36 df-iin 3651 . 2  |^|_  {  |  }
37 df-int 3607 . 2  |^| {  |  }  {  |  {  |  }  }
3835, 36, 373eqtr4g 2094 1  C  |^|_  |^| {  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98  wal 1240   wceq 1242   wcel 1390   {cab 2023  wral 2300  wrex 2301   |^|cint 3606   |^|_ciin 3649
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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  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-int 3607  df-iin 3651
This theorem is referenced by:  dfiin2  3683  iinexgm  3899  dfiin3g  4533  fniinfv  5174
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