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Theorem snriota 6517
Description: A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
Assertion
Ref Expression
snriota  |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  {
( iota_ x  e.  A ph ) } )

Proof of Theorem snriota
StepHypRef Expression
1 df-rab 2659 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-reu 2657 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 sniota 5386 . . . 4  |-  ( E! x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
42, 3sylbi 188 . . 3  |-  ( E! x  e.  A  ph  ->  { x  |  ( x  e.  A  /\  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
51, 4syl5eq 2432 . 2  |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
6 riotaiota 6492 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
76sneqd 3771 . 2  |-  ( E! x  e.  A  ph  ->  { ( iota_ x  e.  A ph ) }  =  { ( iota
x ( x  e.  A  /\  ph )
) } )
85, 7eqtr4d 2423 1  |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  {
( iota_ x  e.  A ph ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   E!weu 2239   {cab 2374   E!wreu 2652   {crab 2654   {csn 3758   iotacio 5357   iota_crio 6479
This theorem is referenced by:  divalgmod  12854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-un 3269  df-if 3684  df-sn 3764  df-pr 3765  df-uni 3959  df-iota 5359  df-riota 6486
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