Theorem riotauni 5729

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 2421	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iotauni 5095	. . 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 5723	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5		df-rab 2423	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	unieqi 3741	. 2 |- U. { x e. A | ph } = U. { x | ( x e. A /\ ph ) }
7	3, 4, 6	3eqtr4g 2195	1 |- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   E!weu 1997   {cab 2123   E!wreu 2416   {crab 2418   U.cuni 3731   iotacio 5081   iota_crio 5722

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-sn 3528  df-pr 3529  df-uni 3732  df-iota 5083  df-riota 5723

This theorem is referenced by:  supval2ti  6875