Theorem rrvfinvima

Description: For a real-value random variable X , any open interval in RR is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017)

	Ref	Expression
Hypotheses	isrrvv.1	|- ( ph -> P e. Prob )
	rrvvf.1	|- ( ph -> X e. ( rRndVar ` P ) )
Assertion	rrvfinvima	|- ( ph -> A. y e. BrSiga ( `' X " y ) e. dom P )

Proof

Step	Hyp	Ref	Expression
1		isrrvv.1	|- ( ph -> P e. Prob )
2		rrvvf.1	|- ( ph -> X e. ( rRndVar ` P ) )
3	1	isrrvv	|- ( ph -> ( X e. ( rRndVar ` P ) <-> ( X : U. dom P --> RR /\ A. y e. BrSiga ( `' X " y ) e. dom P ) ) )
4	2 3	mpbid	|- ( ph -> ( X : U. dom P --> RR /\ A. y e. BrSiga ( `' X " y ) e. dom P ) )
5	4	simprd	|- ( ph -> A. y e. BrSiga ( `' X " y ) e. dom P )