This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: The base set of a constructed uniform space. (Contributed by Thierry
Arnoux, 5-Dec-2017)
|
|
Ref |
Expression |
|
Hypothesis |
tuslem.k |
⊢ 𝐾 = ( toUnifSp ‘ 𝑈 ) |
|
Assertion |
tusbas |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 = ( Base ‘ 𝐾 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
tuslem.k |
⊢ 𝐾 = ( toUnifSp ‘ 𝑈 ) |
| 2 |
1
|
tuslem |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑋 = ( Base ‘ 𝐾 ) ∧ 𝑈 = ( UnifSet ‘ 𝐾 ) ∧ ( unifTop ‘ 𝑈 ) = ( TopOpen ‘ 𝐾 ) ) ) |
| 3 |
2
|
simp1d |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 = ( Base ‘ 𝐾 ) ) |