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Theorem riotacl 5445
Description: Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
Assertion
Ref Expression
riotacl (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotacl
StepHypRef Expression
1 ssrab2 3022 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 riotacl2 5444 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 2940 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1393  ∃!wreu 2305  {crab 2307  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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-sn 3378  df-pr 3379  df-uni 3578  df-iota 4830  df-riota 5431
This theorem is referenced by:  riotaprop  5454  riotass2  5457  riotass  5458  acexmidlemcase  5470  caucvgsrlemcl  6830  caucvgsrlemgt1  6836  axcaucvglemcl  6926  subval  7159  subcl  7166  divvalap  7605  divclap  7609  divfnzn  8504  cjval  9299  cjth  9300  cjf  9301
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