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Theorem riotacl 5839
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 3240 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 riotacl2 5838 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sselid 3153 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  ∃!wreu 2457  {crab 2459  crio 5824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-uni 3808  df-iota 5174  df-riota 5825
This theorem is referenced by:  riotaprop  5848  riotass2  5851  riotass  5852  acexmidlemcase  5864  supclti  6991  caucvgsrlemcl  7776  caucvgsrlemgt1  7782  axcaucvglemcl  7882  subval  8136  subcl  8143  divvalap  8617  divclap  8621  lbcl  8889  divfnzn  9607  flqcl  10256  flapcl  10258  cjval  10835  cjth  10836  cjf  10837  oddpwdclemodd  12152  oddpwdclemdc  12153  oddpwdc  12154  qnumdencl  12167  qnumdenbi  12172  ismgmid  12685  grpinvf  12807
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