Theorem rrvf2

Description: A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017)

	Ref	Expression
Hypotheses	isrrvv.1	⊢ ( 𝜑 → 𝑃 ∈ Prob )
	rrvvf.1	⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
Assertion	rrvf2	⊢ ( 𝜑 → 𝑋 : dom 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	1 2	rrvdm	⊢ ( 𝜑 → dom 𝑋 = ∪ dom 𝑃 )
5	4	feq2d	⊢ ( 𝜑 → ( 𝑋 : dom 𝑋 ⟶ ℝ ↔ 𝑋 : ∪ dom 𝑃 ⟶ ℝ ) )
6	3 5	mpbird	⊢ ( 𝜑 → 𝑋 : dom 𝑋 ⟶ ℝ )