Theorem rrvvf

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	rrvvf	⊢ ( 𝜑 → 𝑋 : ∪ dom 𝑃 ⟶ ℝ )

Proof

Step	Hyp	Ref	Expression
1		isrrvv.1	⊢ ( 𝜑 → 𝑃 ∈ Prob )
2		rrvvf.1	⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
3	1	isrrvv	⊢ ( 𝜑 → ( 𝑋 ∈ ( rRndVar ‘ 𝑃 ) ↔ ( 𝑋 : ∪ dom 𝑃 ⟶ ℝ ∧ ∀ 𝑦 ∈ 𝔅ℝ ( ◡ 𝑋 “ 𝑦 ) ∈ dom 𝑃 ) ) )
4	2 3	mpbid	⊢ ( 𝜑 → ( 𝑋 : ∪ dom 𝑃 ⟶ ℝ ∧ ∀ 𝑦 ∈ 𝔅ℝ ( ◡ 𝑋 “ 𝑦 ) ∈ dom 𝑃 ) )
5	4	simpld	⊢ ( 𝜑 → 𝑋 : ∪ dom 𝑃 ⟶ ℝ )