Theorem snriota 5850

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

Step	Hyp	Ref	Expression
1		df-reu 2460	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		sniota 5199	. . 3 |- ( E! x ( x e. A /\ ph ) -> { x | ( x e. A /\ ph ) } = { ( iota x ( x e. A /\ ph ) ) } )
3	1, 2	sylbi 121	. 2 |- ( E! x e. A ph -> { x | ( x e. A /\ ph ) } = { ( iota x ( x e. A /\ ph ) ) } )
4		df-rab 2462	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-riota 5821	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
6	5	sneqi 3601	. 2 |- { ( iota_ x e. A ph ) } = { ( iota x ( x e. A /\ ph ) ) }
7	3, 4, 6	3eqtr4g 2233	1 |- ( E! x e. A ph -> { x e. A | ph } = { ( iota_ x e. A ph ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E!weu 2024   e. wcel 2146   {cab 2161   E!wreu 2455   {crab 2457   {csn 3589   iotacio 5168   iota_crio 5820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-sn 3595  df-pr 3596  df-uni 3806  df-iota 5170  df-riota 5821

This theorem is referenced by:  divalgmod  11899