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Theorem snriota 5619
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-reu 2366 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 sniota 4994 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
31, 2sylbi 119 . 2  |-  ( E! x  e.  A  ph  ->  { x  |  ( x  e.  A  /\  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
4 df-rab 2368 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
5 df-riota 5590 . . 3  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
65sneqi 3453 . 2  |-  { (
iota_ x  e.  A  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) }
73, 4, 63eqtr4g 2145 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 102    = wceq 1289    e. wcel 1438   E!weu 1948   {cab 2074   E!wreu 2361   {crab 2363   {csn 3441   iotacio 4965   iota_crio 5589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967  df-riota 5590
This theorem is referenced by:  divalgmod  11009
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