Metamath Proof Explorer


Theorem rrvrnss

Description: The range of a random variable as a subset of RR . (Contributed by Thierry Arnoux, 6-Feb-2017)

Ref Expression
Hypotheses isrrvv.1 ( 𝜑𝑃 ∈ Prob )
rrvvf.1 ( 𝜑𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
Assertion rrvrnss ( 𝜑 → ran 𝑋 ⊆ ℝ )

Proof

Step Hyp Ref Expression
1 isrrvv.1 ( 𝜑𝑃 ∈ Prob )
2 rrvvf.1 ( 𝜑𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
3 1 2 rrvvf ( 𝜑𝑋 : dom 𝑃 ⟶ ℝ )
4 3 frnd ( 𝜑 → ran 𝑋 ⊆ ℝ )