Theorem riotauni

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

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

Proof

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