Description: For a real-value random variable X , any open interval in RR is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | isrrvv.1 | ⊢ ( 𝜑 → 𝑃 ∈ Prob ) | |
| rrvvf.1 | ⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) ) | ||
| Assertion | rrvfinvima | ⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝔅ℝ ( ◡ 𝑋 “ 𝑦 ) ∈ dom 𝑃 ) |
| 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 | simprd | ⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝔅ℝ ( ◡ 𝑋 “ 𝑦 ) ∈ dom 𝑃 ) |