Step |
Hyp |
Ref |
Expression |
1 |
|
lpfval.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
1
|
isperf2 |
⊢ ( 𝐽 ∈ Perf ↔ ( 𝐽 ∈ Top ∧ 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) ) |
3 |
|
dfss3 |
⊢ ( 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ∀ 𝑥 ∈ 𝑋 𝑥 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) |
4 |
1
|
maxlp |
⊢ ( 𝐽 ∈ Top → ( 𝑥 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ¬ { 𝑥 } ∈ 𝐽 ) ) ) |
5 |
4
|
baibd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑥 ∈ 𝑋 ) → ( 𝑥 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ¬ { 𝑥 } ∈ 𝐽 ) ) |
6 |
5
|
ralbidva |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑥 ∈ 𝑋 𝑥 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ∀ 𝑥 ∈ 𝑋 ¬ { 𝑥 } ∈ 𝐽 ) ) |
7 |
3 6
|
syl5bb |
⊢ ( 𝐽 ∈ Top → ( 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ∀ 𝑥 ∈ 𝑋 ¬ { 𝑥 } ∈ 𝐽 ) ) |
8 |
7
|
pm5.32i |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝑋 ¬ { 𝑥 } ∈ 𝐽 ) ) |
9 |
2 8
|
bitri |
⊢ ( 𝐽 ∈ Perf ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝑋 ¬ { 𝑥 } ∈ 𝐽 ) ) |