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Theorem snriota 6351
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 2565 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-reu 2563 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 sniota 5262 . . . 4  |-  ( E! x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
42, 3sylbi 187 . . 3  |-  ( E! x  e.  A  ph  ->  { x  |  ( x  e.  A  /\  ph ) }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
51, 4syl5eq 2340 . 2  |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
6 riotaiota 6326 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
76sneqd 3666 . 2  |-  ( E! x  e.  A  ph  ->  { ( iota_ x  e.  A ph ) }  =  { ( iota
x ( x  e.  A  /\  ph )
) } )
85, 7eqtr4d 2331 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 358    = wceq 1632    e. wcel 1696   E!weu 2156   {cab 2282   E!wreu 2558   {crab 2560   {csn 3653   iotacio 5233   iota_crio 6313
This theorem is referenced by:  divalgmod  12621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-un 3170  df-if 3579  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-riota 6320
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