MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotacl Unicode version

Theorem riotacl 6335
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 3271 . 2  |-  { x  e.  A  |  ph }  C_  A
2 riotacl2 6334 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  e.  A  |  ph }
)
31, 2sseldi 3191 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 1696   E!wreu 2558   {crab 2560   iota_crio 6313
This theorem is referenced by:  riotaprop  6344  riotass2  6348  riotass  6349  riotaxfrd  6352  riotaclbg  6360  riotaundb  6362  supcl  7225  fisupcl  7234  htalem  7582  dfac8clem  7675  dfac2a  7772  fin23lem22  7969  zorn2lem1  8139  subcl  9067  divcl  9446  lbcl  9721  flcl  10943  cjf  11605  sqrcl  11861  qnumdencl  12826  qnumdenbi  12831  catidcl  13600  ismgmid  14403  grpinvf  14542  pj1f  15022  grpoidcl  20900  grpoinvcl  20909  iorlid  21011  pjpreeq  21993  cnlnadjlem3  22665  adjbdln  22679  xdivcld  23122  cvmlift3lem3  23867  transportcl  24728  lineval12  26184  lshpkrlem2  29923  lshpkrcl  29928  dihlsscpre  32046  mapdhcl  32539  hdmapcl  32645  hgmapcl  32704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-un 3170  df-in 3172  df-ss 3179  df-if 3579  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-riota 6320
  Copyright terms: Public domain W3C validator