Step |
Hyp |
Ref |
Expression |
1 |
|
orrvccel.1 |
⊢ ( 𝜑 → 𝑃 ∈ Prob ) |
2 |
|
orrvccel.2 |
⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) ) |
3 |
|
orrvccel.4 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
domprobsiga |
⊢ ( 𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra ) |
5 |
1 4
|
syl |
⊢ ( 𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra ) |
6 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( topGen ‘ ran (,) ) ∈ Top ) |
8 |
1
|
rrvmbfm |
⊢ ( 𝜑 → ( 𝑋 ∈ ( rRndVar ‘ 𝑃 ) ↔ 𝑋 ∈ ( dom 𝑃 MblFnM 𝔅ℝ ) ) ) |
9 |
2 8
|
mpbid |
⊢ ( 𝜑 → 𝑋 ∈ ( dom 𝑃 MblFnM 𝔅ℝ ) ) |
10 |
|
df-brsiga |
⊢ 𝔅ℝ = ( sigaGen ‘ ( topGen ‘ ran (,) ) ) |
11 |
10
|
oveq2i |
⊢ ( dom 𝑃 MblFnM 𝔅ℝ ) = ( dom 𝑃 MblFnM ( sigaGen ‘ ( topGen ‘ ran (,) ) ) ) |
12 |
9 11
|
eleqtrdi |
⊢ ( 𝜑 → 𝑋 ∈ ( dom 𝑃 MblFnM ( sigaGen ‘ ( topGen ‘ ran (,) ) ) ) ) |
13 |
5 7 12 3
|
orvcval4 |
⊢ ( 𝜑 → ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) = ( ◡ 𝑋 “ { 𝑦 ∈ ∪ ( topGen ‘ ran (,) ) ∣ 𝑦 𝑅 𝐴 } ) ) |
14 |
|
uniretop |
⊢ ℝ = ∪ ( topGen ‘ ran (,) ) |
15 |
|
rabeq |
⊢ ( ℝ = ∪ ( topGen ‘ ran (,) ) → { 𝑦 ∈ ℝ ∣ 𝑦 𝑅 𝐴 } = { 𝑦 ∈ ∪ ( topGen ‘ ran (,) ) ∣ 𝑦 𝑅 𝐴 } ) |
16 |
14 15
|
ax-mp |
⊢ { 𝑦 ∈ ℝ ∣ 𝑦 𝑅 𝐴 } = { 𝑦 ∈ ∪ ( topGen ‘ ran (,) ) ∣ 𝑦 𝑅 𝐴 } |
17 |
16
|
imaeq2i |
⊢ ( ◡ 𝑋 “ { 𝑦 ∈ ℝ ∣ 𝑦 𝑅 𝐴 } ) = ( ◡ 𝑋 “ { 𝑦 ∈ ∪ ( topGen ‘ ran (,) ) ∣ 𝑦 𝑅 𝐴 } ) |
18 |
13 17
|
eqtr4di |
⊢ ( 𝜑 → ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) = ( ◡ 𝑋 “ { 𝑦 ∈ ℝ ∣ 𝑦 𝑅 𝐴 } ) ) |