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	⊢ ( 𝜑 → 𝑃 ∈ Prob )
	rrvvf.1	⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
Assertion	rrvrnss	⊢ ( 𝜑 → ran 𝑋 ⊆ ℝ )

Proof

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