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Theorem riotacl 5710
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 3150 . 2  |-  { x  e.  A  |  ph }  C_  A
2 riotacl2 5709 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  { x  e.  A  |  ph }
)
31, 2sseldi 3063 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 1463   E!wreu 2393   {crab 2395   iota_crio 5695
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-uni 3705  df-iota 5056  df-riota 5696
This theorem is referenced by:  riotaprop  5719  riotass2  5722  riotass  5723  acexmidlemcase  5735  supclti  6851  caucvgsrlemcl  7561  caucvgsrlemgt1  7567  axcaucvglemcl  7667  subval  7918  subcl  7925  divvalap  8394  divclap  8398  lbcl  8661  divfnzn  9362  flqcl  9986  flapcl  9988  cjval  10557  cjth  10558  cjf  10559  oddpwdclemodd  11745  oddpwdclemdc  11746  oddpwdc  11747  qnumdencl  11760  qnumdenbi  11765
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