Theorem riotauni 5815

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 2455	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iotauni 5172	. . 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 5809	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5		df-rab 2457	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	unieqi 3806	. 2 |- U. { x e. A | ph } = U. { x | ( x e. A /\ ph ) }
7	3, 4, 6	3eqtr4g 2228	1 |- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E!weu 2019   e. wcel 2141   {cab 2156   E!wreu 2450   {crab 2452   U.cuni 3796   iotacio 5158   iota_crio 5808

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160  df-riota 5809

This theorem is referenced by:  supval2ti  6972