Metamath Proof Explorer
Description: Properties that determine a topological space. (Contributed by NM, 20-Oct-2012)
|
|
Ref |
Expression |
|
Hypotheses |
istpsi.b |
⊢ ( Base ‘ 𝐾 ) = 𝐴 |
|
|
istpsi.j |
⊢ ( TopOpen ‘ 𝐾 ) = 𝐽 |
|
|
istpsi.1 |
⊢ 𝐴 = ∪ 𝐽 |
|
|
istpsi.2 |
⊢ 𝐽 ∈ Top |
|
Assertion |
istpsi |
⊢ 𝐾 ∈ TopSp |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
istpsi.b |
⊢ ( Base ‘ 𝐾 ) = 𝐴 |
2 |
|
istpsi.j |
⊢ ( TopOpen ‘ 𝐾 ) = 𝐽 |
3 |
|
istpsi.1 |
⊢ 𝐴 = ∪ 𝐽 |
4 |
|
istpsi.2 |
⊢ 𝐽 ∈ Top |
5 |
1
|
eqcomi |
⊢ 𝐴 = ( Base ‘ 𝐾 ) |
6 |
2
|
eqcomi |
⊢ 𝐽 = ( TopOpen ‘ 𝐾 ) |
7 |
5 6
|
istps2 |
⊢ ( 𝐾 ∈ TopSp ↔ ( 𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽 ) ) |
8 |
4 3 7
|
mpbir2an |
⊢ 𝐾 ∈ TopSp |