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Theorem snriota 6582
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 2716 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-reu 2714 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 sniota 5447 . . . 4  |-  ( E! x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
42, 3sylbi 189 . . 3  |-  ( E! x  e.  A  ph  ->  { x  |  ( x  e.  A  /\  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
51, 4syl5eq 2482 . 2  |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
6 riotaiota 6557 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
76sneqd 3829 . 2  |-  ( E! x  e.  A  ph  ->  { ( iota_ x  e.  A ph ) }  =  { ( iota
x ( x  e.  A  /\  ph )
) } )
85, 7eqtr4d 2473 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 360    = wceq 1653    e. wcel 1726   E!weu 2283   {cab 2424   E!wreu 2709   {crab 2711   {csn 3816   iotacio 5418   iota_crio 6544
This theorem is referenced by:  divalgmod  12928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-un 3327  df-if 3742  df-sn 3822  df-pr 3823  df-uni 4018  df-iota 5420  df-riota 6551
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