Theorem snriota 5903

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 2479	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		sniota 5245	. . 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 2481	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-riota 5873	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
6	5	sneqi 3630	. 2 |- { ( iota_ x e. A ph ) } = { ( iota x ( x e. A /\ ph ) ) }
7	3, 4, 6	3eqtr4g 2251	1 |- ( E! x e. A ph -> { x e. A | ph } = { ( iota_ x e. A ph ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E!weu 2042   e. wcel 2164   {cab 2179   E!wreu 2474   {crab 2476   {csn 3618   iotacio 5213   iota_crio 5872

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-sn 3624  df-pr 3625  df-uni 3836  df-iota 5215  df-riota 5873

This theorem is referenced by:  divalgmod  12068