Theorem elorrvc

Description: Elementhood of a preimage for a real-valued random variable. (Contributed by Thierry Arnoux, 21-Jan-2017)

	Ref	Expression
Hypotheses	orrvccel.1	⊢ ( 𝜑 → 𝑃 ∈ Prob )
	orrvccel.2	⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
	orrvccel.4	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
Assertion	elorrvc	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃 ) → ( 𝑧 ∈ ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) ↔ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		orrvccel.1	⊢ ( 𝜑 → 𝑃 ∈ Prob )
2		orrvccel.2	⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) )
3		orrvccel.4	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
4	1 2	rrvdm	⊢ ( 𝜑 → dom 𝑋 = ∪ dom 𝑃 )
5	4	eleq2d	⊢ ( 𝜑 → ( 𝑧 ∈ dom 𝑋 ↔ 𝑧 ∈ ∪ dom 𝑃 ) )
6	5	biimprd	⊢ ( 𝜑 → ( 𝑧 ∈ ∪ dom 𝑃 → 𝑧 ∈ dom 𝑋 ) )
7	6	imdistani	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃 ) → ( 𝜑 ∧ 𝑧 ∈ dom 𝑋 ) )
8	1 2	rrvfn	⊢ ( 𝜑 → 𝑋 Fn ∪ dom 𝑃 )
9		fnfun	⊢ ( 𝑋 Fn ∪ dom 𝑃 → Fun 𝑋 )
10	8 9	syl	⊢ ( 𝜑 → Fun 𝑋 )
11	10 2 3	elorvc	⊢ ( ( 𝜑 ∧ 𝑧 ∈ dom 𝑋 ) → ( 𝑧 ∈ ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) ↔ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 ) )
12	7 11	syl	⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃 ) → ( 𝑧 ∈ ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) ↔ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 ) )