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	$$⊢\phi \to P\in \mathrm{Prob}$$
	rrvvf.1	$$⊢\phi \to X\in {\mathrm{RndVar}}_{ℝ}\left(P\right)$$
Assertion	rrvrnss	$$⊢\phi \to \mathrm{ran}X\subseteq ℝ$$

Proof

Step	Hyp	Ref	Expression
1		isrrvv.1	$$⊢\phi \to P\in \mathrm{Prob}$$
2		rrvvf.1	$$⊢\phi \to X\in {\mathrm{RndVar}}_{ℝ}\left(P\right)$$
3	1 2	rrvvf	$$⊢\phi \to X:\bigcup \mathrm{dom}P⟶ℝ$$
4	3	frnd	$$⊢\phi \to \mathrm{ran}X\subseteq ℝ$$