Theorem snriota 6070

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 2535	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		sniota 5368	. . 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 2537	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-riota 6038	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
6	5	sneqi 3721	. 2 |- { ( iota_ x e. A ph ) } = { ( iota x ( x e. A /\ ph ) ) }
7	3, 4, 6	3eqtr4g 2296	1 |- ( E! x e. A ph -> { x e. A | ph } = { ( iota_ x e. A ph ) } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E!weu 2086   e. wcel 2209   {cab 2224   E!wreu 2530   {crab 2532   {csn 3709   iotacio 5335   iota_crio 6037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3715  df-pr 3716  df-uni 3936  df-iota 5337  df-riota 6038

This theorem is used by:  divalgmod  12694