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Theorem riotacl 5823
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 3232 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 riotacl2 5822 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sselid 3145 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2141  ∃!wreu 2450  {crab 2452  crio 5808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160  df-riota 5809
This theorem is referenced by:  riotaprop  5832  riotass2  5835  riotass  5836  acexmidlemcase  5848  supclti  6975  caucvgsrlemcl  7751  caucvgsrlemgt1  7757  axcaucvglemcl  7857  subval  8111  subcl  8118  divvalap  8591  divclap  8595  lbcl  8862  divfnzn  9580  flqcl  10229  flapcl  10231  cjval  10809  cjth  10810  cjf  10811  oddpwdclemodd  12126  oddpwdclemdc  12127  oddpwdc  12128  qnumdencl  12141  qnumdenbi  12146  ismgmid  12631  grpinvf  12750
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