Metamath Proof Explorer


Theorem isperf2

Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016)

Ref Expression
Hypothesis lpfval.1 𝑋 = 𝐽
Assertion isperf2 ( 𝐽 ∈ Perf ↔ ( 𝐽 ∈ Top ∧ 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 lpfval.1 𝑋 = 𝐽
2 1 isperf ( 𝐽 ∈ Perf ↔ ( 𝐽 ∈ Top ∧ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) = 𝑋 ) )
3 ssid 𝑋𝑋
4 1 lpss ( ( 𝐽 ∈ Top ∧ 𝑋𝑋 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ⊆ 𝑋 )
5 3 4 mpan2 ( 𝐽 ∈ Top → ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ⊆ 𝑋 )
6 eqss ( ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) = 𝑋 ↔ ( ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ⊆ 𝑋𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) )
7 6 baib ( ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ⊆ 𝑋 → ( ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) = 𝑋𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) )
8 5 7 syl ( 𝐽 ∈ Top → ( ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) = 𝑋𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) )
9 8 pm5.32i ( ( 𝐽 ∈ Top ∧ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) = 𝑋 ) ↔ ( 𝐽 ∈ Top ∧ 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) )
10 2 9 bitri ( 𝐽 ∈ Perf ↔ ( 𝐽 ∈ Top ∧ 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) )