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Theorem riotaexg 5500
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 5496 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
2 uniexg 4203 . . 3 (𝐴𝑉 𝐴 ∈ V)
3 iotass 4912 . . . . 5 (∀𝑥((𝑥𝐴𝜓) → 𝑥 𝐴) → (℩𝑥(𝑥𝐴𝜓)) ⊆ 𝐴)
4 elssuni 3636 . . . . . 6 (𝑥𝐴𝑥 𝐴)
54adantr 265 . . . . 5 ((𝑥𝐴𝜓) → 𝑥 𝐴)
63, 5mpg 1356 . . . 4 (℩𝑥(𝑥𝐴𝜓)) ⊆ 𝐴
76a1i 9 . . 3 (𝐴𝑉 → (℩𝑥(𝑥𝐴𝜓)) ⊆ 𝐴)
82, 7ssexd 3925 . 2 (𝐴𝑉 → (℩𝑥(𝑥𝐴𝜓)) ∈ V)
91, 8syl5eqel 2140 1 (𝐴𝑉 → (𝑥𝐴 𝜓) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  Vcvv 2574  wss 2945   cuni 3608  cio 4893  crio 5495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-iota 4895  df-riota 5496
This theorem is referenced by:  flval  9224  sqrtrval  9827
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