Theorem riotauni 5688

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 2395	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iotauni 5056	. . 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 5682	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5		df-rab 2397	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	unieqi 3710	. 2 |- U. { x e. A | ph } = U. { x | ( x e. A /\ ph ) }
7	3, 4, 6	3eqtr4g 2170	1 |- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   E!weu 1973   {cab 2099   E!wreu 2390   {crab 2392   U.cuni 3700   iotacio 5042   iota_crio 5681

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-un 3039  df-sn 3497  df-pr 3498  df-uni 3701  df-iota 5044  df-riota 5682

This theorem is referenced by:  supval2ti  6832