Step |
Hyp |
Ref |
Expression |
1 |
|
ressust.x |
⊢ 𝑋 = ( Base ‘ 𝑊 ) |
2 |
|
ressust.t |
⊢ 𝑇 = ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) |
3 |
1
|
fvexi |
⊢ 𝑋 ∈ V |
4 |
3
|
ssex |
⊢ ( 𝐴 ⊆ 𝑋 → 𝐴 ∈ V ) |
5 |
4
|
adantl |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋 ) → 𝐴 ∈ V ) |
6 |
|
ressuss |
⊢ ( 𝐴 ∈ V → ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) = ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ) |
7 |
5 6
|
syl |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋 ) → ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) = ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ) |
8 |
2 7
|
eqtrid |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋 ) → 𝑇 = ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ) |
9 |
|
eqid |
⊢ ( UnifSt ‘ 𝑊 ) = ( UnifSt ‘ 𝑊 ) |
10 |
|
eqid |
⊢ ( TopOpen ‘ 𝑊 ) = ( TopOpen ‘ 𝑊 ) |
11 |
1 9 10
|
isusp |
⊢ ( 𝑊 ∈ UnifSp ↔ ( ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝑋 ) ∧ ( TopOpen ‘ 𝑊 ) = ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ) ) |
12 |
11
|
simplbi |
⊢ ( 𝑊 ∈ UnifSp → ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝑋 ) ) |
13 |
|
trust |
⊢ ( ( ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) ) |
14 |
12 13
|
sylan |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋 ) → ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) ) |
15 |
8 14
|
eqeltrd |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋 ) → 𝑇 ∈ ( UnifOn ‘ 𝐴 ) ) |