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Theorem riotaexg 5415
Description: Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
Assertion
Ref Expression
riotaexg (A 𝑉 → (x A ψ) V)
Distinct variable group:   x,A
Allowed substitution hints:   ψ(x)   𝑉(x)

Proof of Theorem riotaexg
StepHypRef Expression
1 df-riota 5411 . 2 (x A ψ) = (℩x(x A ψ))
2 uniexg 4141 . . 3 (A 𝑉 A V)
3 iotass 4827 . . . . 5 (x((x A ψ) → x A) → (℩x(x A ψ)) ⊆ A)
4 elssuni 3599 . . . . . 6 (x Ax A)
54adantr 261 . . . . 5 ((x A ψ) → x A)
63, 5mpg 1337 . . . 4 (℩x(x A ψ)) ⊆ A
76a1i 9 . . 3 (A 𝑉 → (℩x(x A ψ)) ⊆ A)
82, 7ssexd 3888 . 2 (A 𝑉 → (℩x(x A ψ)) V)
91, 8syl5eqel 2121 1 (A 𝑉 → (x A ψ) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  wss 2911   cuni 3571  cio 4808  crio 5410
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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136
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  df-riota 5411
This theorem is referenced by:  sqrtrval  9209
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