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Theorem snriota 6539
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 2675 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-reu 2673 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 sniota 5404 . . . 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 2448 . 2  |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
6 riotaiota 6514 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
76sneqd 3787 . 2  |-  ( E! x  e.  A  ph  ->  { ( iota_ x  e.  A ph ) }  =  { ( iota
x ( x  e.  A  /\  ph )
) } )
85, 7eqtr4d 2439 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 1721   E!weu 2254   {cab 2390   E!wreu 2668   {crab 2670   {csn 3774   iotacio 5375   iota_crio 6501
This theorem is referenced by:  divalgmod  12881
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-un 3285  df-if 3700  df-sn 3780  df-pr 3781  df-uni 3976  df-iota 5377  df-riota 6508
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