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Theorem riotaexg 5741
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 5737 . 2  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
2 uniexg 4368 . . 3  |-  ( A  e.  V  ->  U. A  e.  _V )
3 iotass 5112 . . . . 5  |-  ( A. x ( ( x  e.  A  /\  ps )  ->  x  C_  U. A
)  ->  ( iota x ( x  e.  A  /\  ps )
)  C_  U. A )
4 elssuni 3771 . . . . . 6  |-  ( x  e.  A  ->  x  C_ 
U. A )
54adantr 274 . . . . 5  |-  ( ( x  e.  A  /\  ps )  ->  x  C_  U. A )
63, 5mpg 1428 . . . 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 4075 . 2  |-  ( A  e.  V  ->  ( iota x ( x  e.  A  /\  ps )
)  e.  _V )
91, 8eqeltrid 2227 1  |-  ( A  e.  V  ->  ( iota_ x  e.  A  ps )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1481   _Vcvv 2689    C_ wss 3075   U.cuni 3743   iotacio 5093   iota_crio 5736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-uni 3744  df-iota 5095  df-riota 5737
This theorem is referenced by:  flval  10075  sqrtrval  10803  qnumval  11897  qdenval  11898
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