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Theorem riotacl 5564
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 3092 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 riotacl2 5563 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 3010 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1436  ∃!wreu 2357  {crab 2359  crio 5549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-sn 3431  df-pr 3432  df-uni 3631  df-iota 4937  df-riota 5550
This theorem is referenced by:  riotaprop  5573  riotass2  5576  riotass  5577  acexmidlemcase  5589  supclti  6614  caucvgsrlemcl  7255  caucvgsrlemgt1  7261  axcaucvglemcl  7351  subval  7595  subcl  7602  divvalap  8057  divclap  8061  lbcl  8319  divfnzn  9015  flqcl  9583  flapcl  9585  cjval  10120  cjth  10121  cjf  10122  oddpwdclemodd  10944  oddpwdclemdc  10945  oddpwdc  10946  qnumdencl  10959  qnumdenbi  10964
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