ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  riotacl GIF version

Theorem riotacl 5744
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 3182 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
2 riotacl2 5743 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ {𝑥𝐴𝜑})
31, 2sseldi 3095 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480  ∃!wreu 2418  {crab 2420  crio 5729
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088  df-riota 5730
This theorem is referenced by:  riotaprop  5753  riotass2  5756  riotass  5757  acexmidlemcase  5769  supclti  6885  caucvgsrlemcl  7609  caucvgsrlemgt1  7615  axcaucvglemcl  7715  subval  7966  subcl  7973  divvalap  8446  divclap  8450  lbcl  8716  divfnzn  9425  flqcl  10058  flapcl  10060  cjval  10629  cjth  10630  cjf  10631  oddpwdclemodd  11861  oddpwdclemdc  11862  oddpwdc  11863  qnumdencl  11876  qnumdenbi  11881
  Copyright terms: Public domain W3C validator