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Theorem iinexgm 3899
Description: The existence of an indexed union. is normally a free-variable parameter in , which should be read . (Contributed by Jim Kingdon, 28-Aug-2018.)
Assertion
Ref Expression
iinexgm  C  |^|_  _V
Distinct variable group:   ,
Allowed substitution hints:   ()    C()

Proof of Theorem iinexgm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfiin2g 3681 . . 3  C  |^|_  |^| {  |  }
21adantl 262 . 2  C  |^|_  |^| {  |  }
3 elisset 2562 . . . . . . . . . 10  C
43rgenw 2370 . . . . . . . . 9  C
5 r19.2m 3303 . . . . . . . . 9  C  C
64, 5mpan2 401 . . . . . . . 8  C
7 r19.35-1 2454 . . . . . . . 8  C  C
86, 7syl 14 . . . . . . 7  C
98imp 115 . . . . . 6  C
10 rexcom4 2571 . . . . . 6
119, 10sylib 127 . . . . 5  C
12 abid 2025 . . . . . 6  {  |  }
1312exbii 1493 . . . . 5  {  |  }
1411, 13sylibr 137 . . . 4  C  {  |  }
15 nfv 1418 . . . . 5  F/  {  |  }
16 nfsab1 2027 . . . . 5  F/  {  |  }
17 eleq1 2097 . . . . 5  {  |  }  {  |  }
1815, 16, 17cbvex 1636 . . . 4  {  |  } 
{  |  }
1914, 18sylib 127 . . 3  C  {  |  }
20 inteximm 3894 . . 3  {  |  }  |^| {  |  }  _V
2119, 20syl 14 . 2  C  |^| {  |  }  _V
222, 21eqeltrd 2111 1  C  |^|_  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   {cab 2023  wral 2300  wrex 2301   _Vcvv 2551   |^|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  ax-sep 3866
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-in 2918  df-ss 2925  df-int 3607  df-iin 3651
This theorem is referenced by: (None)
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