Theorem rrvf2

Description: A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-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	rrvf2	$$⊢\phi \to X:\mathrm{dom}X⟶ℝ$$

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	1 2	rrvdm	$$⊢\phi \to \mathrm{dom}X=\bigcup \mathrm{dom}P$$
5	4	feq2d	$$⊢\phi \to \left(X:\mathrm{dom}X⟶ℝ\leftrightarrow X:\bigcup \mathrm{dom}P⟶ℝ\right)$$
6	3 5	mpbird	$$⊢\phi \to X:\mathrm{dom}X⟶ℝ$$