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Theorem riotacl2 5750
Description: Membership law for "the unique element in  A such that  ph."

(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 df-reu 2424 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iotacl 5118 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  |  ( x  e.  A  /\  ph ) } )
31, 2sylbi 120 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  |  ( x  e.  A  /\  ph ) } )
4 df-riota 5737 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
5 df-rab 2426 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
63, 4, 53eltr4g 2226 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 103    e. wcel 1481   E!weu 2000   {cab 2126   E!wreu 2419   {crab 2421   iotacio 5093   iota_crio 5736
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-un 3079  df-sn 3537  df-pr 3538  df-uni 3744  df-iota 5095  df-riota 5737
This theorem is referenced by:  riotacl  5751  riotasbc  5752  supubti  6893  suplubti  6894
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