Step |
Hyp |
Ref |
Expression |
1 |
|
hoissrrn2.kph |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
hoissrrn2.a |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐴 ∈ ℝ ) |
3 |
|
hoissrrn2.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐵 ∈ ℝ* ) |
4 |
|
ovex |
⊢ ( 𝐴 [,) 𝐵 ) ∈ V |
5 |
4
|
rgenw |
⊢ ∀ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ∈ V |
6 |
|
ixpssmapg |
⊢ ( ∀ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ∈ V → X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( ∪ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ) |
7 |
5 6
|
ax-mp |
⊢ X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( ∪ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( ∪ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ) |
9 |
|
reex |
⊢ ℝ ∈ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
11 |
|
icossre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) |
12 |
2 3 11
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) |
13 |
12
|
ex |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 → ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) ) |
14 |
1 13
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) |
15 |
|
iunss |
⊢ ( ∪ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ ↔ ∀ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) |
16 |
14 15
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) |
17 |
|
mapss |
⊢ ( ( ℝ ∈ V ∧ ∪ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ℝ ) → ( ∪ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
18 |
10 16 17
|
syl2anc |
⊢ ( 𝜑 → ( ∪ 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
19 |
8 18
|
sstrd |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( 𝐴 [,) 𝐵 ) ⊆ ( ℝ ↑m 𝑋 ) ) |