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Theorem snriota 6335
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 2552 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-reu 2550 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 sniota 5246 . . . 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 2327 . 2  |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  {
( iota x ( x  e.  A  /\  ph ) ) } )
6 riotaiota 6310 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
76sneqd 3653 . 2  |-  ( E! x  e.  A  ph  ->  { ( iota_ x  e.  A ph ) }  =  { ( iota
x ( x  e.  A  /\  ph )
) } )
85, 7eqtr4d 2318 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 1623    e. wcel 1684   E!weu 2143   {cab 2269   E!wreu 2545   {crab 2547   {csn 3640   iotacio 5217   iota_crio 6297
This theorem is referenced by:  divalgmod  12605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-un 3157  df-if 3566  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-riota 6304
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