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Theorem riotaexg 5523
Description: Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
Assertion
Ref Expression
riotaexg  |-  ( A  e.  V  ->  ( iota_ x  e.  A  ps )  e.  _V )
Distinct variable group:    x, A
Allowed substitution hints:    ps( x)    V( x)

Proof of Theorem riotaexg
StepHypRef Expression
1 df-riota 5519 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
2 uniexg 4221 . . 3  |-  ( A  e.  V  ->  U. A  e.  _V )
3 iotass 4934 . . . . 5  |-  ( A. x ( ( x  e.  A  /\  ps )  ->  x  C_  U. A
)  ->  ( iota x ( x  e.  A  /\  ps )
)  C_  U. A )
4 elssuni 3649 . . . . . 6  |-  ( x  e.  A  ->  x  C_ 
U. A )
54adantr 270 . . . . 5  |-  ( ( x  e.  A  /\  ps )  ->  x  C_  U. A )
63, 5mpg 1381 . . . 4  |-  ( iota
x ( x  e.  A  /\  ps )
)  C_  U. A
76a1i 9 . . 3  |-  ( A  e.  V  ->  ( iota x ( x  e.  A  /\  ps )
)  C_  U. A )
82, 7ssexd 3938 . 2  |-  ( A  e.  V  ->  ( iota x ( x  e.  A  /\  ps )
)  e.  _V )
91, 8syl5eqel 2169 1  |-  ( A  e.  V  ->  ( iota_ x  e.  A  ps )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   _Vcvv 2610    C_ wss 2982   U.cuni 3621   iotacio 4915   iota_crio 5518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-iota 4917  df-riota 5519
This theorem is referenced by:  flval  9406  sqrtrval  10087  qnumval  10770  qdenval  10771
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