Metamath Proof Explorer


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 φ P Prob
rrvvf.1 φ X RndVar P
Assertion rrvfinvima φ y 𝔅 X -1 y dom P

Proof

Step Hyp Ref Expression
1 isrrvv.1 φ P Prob
2 rrvvf.1 φ X RndVar P
3 1 isrrvv φ X RndVar P X : dom P y 𝔅 X -1 y dom P
4 2 3 mpbid φ X : dom P y 𝔅 X -1 y dom P
5 4 simprd φ y 𝔅 X -1 y dom P