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Theorem riotaexg 5435
Description: Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
Assertion
Ref Expression
riotaexg (𝐴𝑉 → (𝑥𝐴 𝜓) ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem riotaexg
StepHypRef Expression
1 df-riota 5431 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
2 uniexg 4147 . . 3 (𝐴𝑉 𝐴 ∈ V)
3 iotass 4847 . . . . 5 (∀𝑥((𝑥𝐴𝜓) → 𝑥 𝐴) → (℩𝑥(𝑥𝐴𝜓)) ⊆ 𝐴)
4 elssuni 3605 . . . . . 6 (𝑥𝐴𝑥 𝐴)
54adantr 261 . . . . 5 ((𝑥𝐴𝜓) → 𝑥 𝐴)
63, 5mpg 1340 . . . 4 (℩𝑥(𝑥𝐴𝜓)) ⊆ 𝐴
76a1i 9 . . 3 (𝐴𝑉 → (℩𝑥(𝑥𝐴𝜓)) ⊆ 𝐴)
82, 7ssexd 3894 . 2 (𝐴𝑉 → (℩𝑥(𝑥𝐴𝜓)) ∈ V)
91, 8syl5eqel 2124 1 (𝐴𝑉 → (𝑥𝐴 𝜓) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wcel 1393  Vcvv 2554  wss 2914   cuni 3577  cio 4828  crio 5430
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-un 4142
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-uni 3578  df-iota 4830  df-riota 5431
This theorem is referenced by:  sqrtrval  9452
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