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Theorem riotacl2 5843
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 2462 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iotacl 5201 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  |  ( x  e.  A  /\  ph ) } )
31, 2sylbi 121 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  e.  {
x  |  ( x  e.  A  /\  ph ) } )
4 df-riota 5830 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
5 df-rab 2464 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
63, 4, 53eltr4g 2263 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 104   E!weu 2026    e. wcel 2148   {cab 2163   E!wreu 2457   {crab 2459   iotacio 5176   iota_crio 5829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3598  df-pr 3599  df-uni 3810  df-iota 5178  df-riota 5830
This theorem is referenced by:  riotacl  5844  riotasbc  5845  supubti  6997  suplubti  6998  grplinv  12876
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