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Theorem riotacl2 6334
Description: Membership law for "the unique element in  A such that  ph."

This can useful for expanding an iota-based definition (see df-iota 5235). If you have an unbounded iota, iotacl 5258 may be useful.

(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion
Ref Expression
riotacl2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  e.  A  |  ph }
)

Proof of Theorem riotacl2
StepHypRef Expression
1 riotaiota 6326 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
2 df-reu 2563 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iotacl 5258 . . . 4  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  |  ( x  e.  A  /\  ph ) } )
42, 3sylbi 187 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  |  ( x  e.  A  /\  ph ) } )
5 df-rab 2565 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
64, 5syl6eleqr 2387 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  e.  A  |  ph } )
71, 6eqeltrd 2370 1  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  { x  e.  A  |  ph }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   E!weu 2156   {cab 2282   E!wreu 2558   {crab 2560   iotacio 5233   iota_crio 6313
This theorem is referenced by:  riotacl  6335  riotasbc  6336  riotaxfrd  6352  supub  7226  suplub  7227  ordtypelem3  7251  catlid  13601  catrid  13602  grplinv  14544  pj1id  15024  evlsval2  19420  ig1pval3  19576  coelem  19624  quotlem  19696  grpoidinv2  20901  grpoinv  20910  cnlnadjlem5  22667  cvmsiota  23823  cvmliftiota  23847  supdef  25365  mpaalem  27460
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-if 3579  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-riota 6320
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