Theorem rrvdm

Description: The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017)

	Ref	Expression
Hypotheses	isrrvv.1	⊢ ( 𝜑 → 𝑃 ∈ Prob )
	rrvvf.1	⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
Assertion	rrvdm	⊢ ( 𝜑 → dom 𝑋 = ∪ dom 𝑃 )

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	fdmd	⊢ ( 𝜑 → dom 𝑋 = ∪ dom 𝑃 )