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Theorem riotacl 5513
Description: Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
Assertion
Ref Expression
riotacl  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem riotacl
StepHypRef Expression
1 ssrab2 3080 . 2  |-  { x  e.  A  |  ph }  C_  A
2 riotacl2 5512 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  e.  A  |  ph }
)
31, 2sseldi 2998 1  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   E!wreu 2351   {crab 2353   iota_crio 5498
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-uni 3610  df-iota 4897  df-riota 5499
This theorem is referenced by:  riotaprop  5522  riotass2  5525  riotass  5526  acexmidlemcase  5538  supclti  6470  caucvgsrlemcl  7027  caucvgsrlemgt1  7033  axcaucvglemcl  7123  subval  7367  subcl  7374  divvalap  7829  divclap  7833  lbcl  8091  divfnzn  8787  flqcl  9355  flapcl  9357  cjval  9870  cjth  9871  cjf  9872  oddpwdclemodd  10694  oddpwdclemdc  10695  oddpwdc  10696
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