Theorem snriota 5619

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 2366	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		sniota 4994	. . 3 |- ( E! x ( x e. A /\ ph ) -> { x | ( x e. A /\ ph ) } = { ( iota x ( x e. A /\ ph ) ) } )
3	1, 2	sylbi 119	. 2 |- ( E! x e. A ph -> { x | ( x e. A /\ ph ) } = { ( iota x ( x e. A /\ ph ) ) } )
4		df-rab 2368	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-riota 5590	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
6	5	sneqi 3453	. 2 |- { ( iota_ x e. A ph ) } = { ( iota x ( x e. A /\ ph ) ) }
7	3, 4, 6	3eqtr4g 2145	1 |- ( E! x e. A ph -> { x e. A | ph } = { ( iota_ x e. A ph ) } )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   E!weu 1948   {cab 2074   E!wreu 2361   {crab 2363   {csn 3441   iotacio 4965   iota_crio 5589

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967  df-riota 5590

This theorem is referenced by:  divalgmod  11009