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 ) ) |