Theorem rrvdmss

Description: The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		isrrvv.1	⊢ ( 𝜑 → 𝑃 ∈ Prob )
2		rrvvf.1	⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
3	1 2	rrvdm	⊢ ( 𝜑 → dom 𝑋 = ∪ dom 𝑃 )
4		eqimss2	⊢ ( dom 𝑋 = ∪ dom 𝑃 → ∪ dom 𝑃 ⊆ dom 𝑋 )
5	3 4	syl	⊢ ( 𝜑 → ∪ dom 𝑃 ⊆ dom 𝑋 )