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Theorem iotass 4827
Description: Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
Assertion
Ref Expression
iotass  C_  iota  C_
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem iotass
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-iota 4810 . 2  iota  U. {  |  {  |  }  { } }
2 unieq 3580 . . . . . . . 8  {  |  }  { }  U. {  |  }  U. { }
3 vex 2554 . . . . . . . . 9 
_V
43unisn 3587 . . . . . . . 8  U. { }
52, 4syl6eq 2085 . . . . . . 7  {  |  }  { }  U. {  |  }
6 df-pw 3353 . . . . . . . . . . 11  ~P  {  | 
C_  }
76sseq2i 2964 . . . . . . . . . 10  {  |  }  C_  ~P  {  |  }  C_  {  |  C_  }
8 ss2ab 3002 . . . . . . . . . 10  {  |  }  C_  {  |  C_  }  C_
97, 8bitri 173 . . . . . . . . 9  {  |  }  C_  ~P  C_
109biimpri 124 . . . . . . . 8  C_  {  |  }  C_  ~P
11 sspwuni 3730 . . . . . . . 8  {  |  }  C_  ~P  U. {  |  }  C_
1210, 11sylib 127 . . . . . . 7  C_  U. {  |  }  C_
13 sseq1 2960 . . . . . . . 8  U. {  |  }  U. {  |  }  C_  C_
1413biimpa 280 . . . . . . 7 
U. {  |  }  U. {  |  }  C_  C_
155, 12, 14syl2anr 274 . . . . . 6 
C_  {  |  }  { }  C_
1615ex 108 . . . . 5  C_  {  |  }  { }  C_
1716ss2abdv 3007 . . . 4  C_  {  |  {  |  }  { } }  C_  {  |  C_  }
18 df-pw 3353 . . . 4  ~P  {  | 
C_  }
1917, 18syl6sseqr 2986 . . 3  C_  {  |  {  |  }  { } }  C_  ~P
20 sspwuni 3730 . . 3  {  |  {  |  }  { } }  C_  ~P  U. {  |  {  |  }  { } }  C_
2119, 20sylib 127 . 2  C_  U. {  |  {  |  }  { } }  C_
221, 21syl5eqss 2983 1  C_  iota  C_
Colors of variables: wff set class
Syntax hints:   wi 4  wal 1240   wceq 1242   {cab 2023    C_ wss 2911   ~Pcpw 3351   {csn 3367   U.cuni 3571   iotacio 4808
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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810
This theorem is referenced by:  fvss  5132  riotaexg  5415
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