Theorem riotauni 5804

Description: Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)

Assertion

Ref	Expression
riotauni	|- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )

Proof of Theorem riotauni

Step	Hyp	Ref	Expression
1		df-reu 2451	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iotauni 5165	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) = U. { x | ( x e. A /\ ph ) } )
3	1, 2	sylbi 120	. 2 |- ( E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) = U. { x | ( x e. A /\ ph ) } )
4		df-riota 5798	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5		df-rab 2453	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	unieqi 3799	. 2 |- U. { x e. A | ph } = U. { x | ( x e. A /\ ph ) }
7	3, 4, 6	3eqtr4g 2224	1 |- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E!weu 2014   e. wcel 2136   {cab 2151   E!wreu 2446   {crab 2448   U.cuni 3789   iotacio 5151   iota_crio 5797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153  df-riota 5798

This theorem is referenced by:  supval2ti  6960