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Theorem riotacl 5510
 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 3053 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 riotacl2 5509 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 2971 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1409  ∃!wreu 2325  {crab 2327  ℩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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-uni 3609  df-iota 4895  df-riota 5496 This theorem is referenced by:  riotaprop  5519  riotass2  5522  riotass  5523  acexmidlemcase  5535  supclti  6404  caucvgsrlemcl  6931  caucvgsrlemgt1  6937  axcaucvglemcl  7027  subval  7266  subcl  7273  divvalap  7727  divclap  7731  divfnzn  8653  flqcl  9225  cjval  9673  cjth  9674  cjf  9675  oddpwdclemodd  10260  oddpwdclemdc  10261  oddpwdc  10262
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