Step |
Hyp |
Ref |
Expression |
1 |
|
dstfrv.1 |
⊢ ( 𝜑 → 𝑃 ∈ Prob ) |
2 |
|
dstfrv.2 |
⊢ ( 𝜑 → 𝑋 ∈ ( rRndVar ‘ 𝑃 ) ) |
3 |
|
orvclteel.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
4 |
|
dstfrvel.1 |
⊢ ( 𝜑 → 𝐵 ∈ ∪ dom 𝑃 ) |
5 |
|
dstfrvel.2 |
⊢ ( 𝜑 → ( 𝑋 ‘ 𝐵 ) ≤ 𝐴 ) |
6 |
1 2
|
rrvvf |
⊢ ( 𝜑 → 𝑋 : ∪ dom 𝑃 ⟶ ℝ ) |
7 |
6 4
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑋 ‘ 𝐵 ) ∈ ℝ ) |
8 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑋 ‘ 𝐵 ) → ( 𝑥 ≤ 𝐴 ↔ ( 𝑋 ‘ 𝐵 ) ≤ 𝐴 ) ) |
9 |
8
|
elrab |
⊢ ( ( 𝑋 ‘ 𝐵 ) ∈ { 𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴 } ↔ ( ( 𝑋 ‘ 𝐵 ) ∈ ℝ ∧ ( 𝑋 ‘ 𝐵 ) ≤ 𝐴 ) ) |
10 |
7 5 9
|
sylanbrc |
⊢ ( 𝜑 → ( 𝑋 ‘ 𝐵 ) ∈ { 𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴 } ) |
11 |
6
|
ffund |
⊢ ( 𝜑 → Fun 𝑋 ) |
12 |
1 2
|
rrvdm |
⊢ ( 𝜑 → dom 𝑋 = ∪ dom 𝑃 ) |
13 |
4 12
|
eleqtrrd |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝑋 ) |
14 |
|
fvimacnv |
⊢ ( ( Fun 𝑋 ∧ 𝐵 ∈ dom 𝑋 ) → ( ( 𝑋 ‘ 𝐵 ) ∈ { 𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴 } ↔ 𝐵 ∈ ( ◡ 𝑋 “ { 𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴 } ) ) ) |
15 |
11 13 14
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑋 ‘ 𝐵 ) ∈ { 𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴 } ↔ 𝐵 ∈ ( ◡ 𝑋 “ { 𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴 } ) ) ) |
16 |
10 15
|
mpbid |
⊢ ( 𝜑 → 𝐵 ∈ ( ◡ 𝑋 “ { 𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴 } ) ) |
17 |
1 2 3
|
orrvcval4 |
⊢ ( 𝜑 → ( 𝑋 ∘RV/𝑐 ≤ 𝐴 ) = ( ◡ 𝑋 “ { 𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴 } ) ) |
18 |
16 17
|
eleqtrrd |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝑋 ∘RV/𝑐 ≤ 𝐴 ) ) |