Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
2 |
1
|
fclselbas |
⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐴 ∈ ∪ 𝐽 ) |
3 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
4 |
3
|
adantr |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → 𝑋 = ∪ 𝐽 ) |
5 |
4
|
eleq2d |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ 𝐽 ) ) |
6 |
2 5
|
syl5ibr |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐴 ∈ 𝑋 ) ) |
7 |
|
fclsneii |
⊢ ( ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ∧ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∧ 𝑠 ∈ 𝐹 ) → ( 𝑛 ∩ 𝑠 ) ≠ ∅ ) |
8 |
7
|
3expb |
⊢ ( ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ∧ ( 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∧ 𝑠 ∈ 𝐹 ) ) → ( 𝑛 ∩ 𝑠 ) ≠ ∅ ) |
9 |
8
|
ralrimivva |
⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ ) |
10 |
6 9
|
jca2 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ ) ) ) |
11 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
12 |
11
|
ad3antrrr |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜 ) ) → 𝐽 ∈ Top ) |
13 |
|
simprl |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜 ) ) → 𝑜 ∈ 𝐽 ) |
14 |
|
simprr |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜 ) ) → 𝐴 ∈ 𝑜 ) |
15 |
|
opnneip |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜 ) → 𝑜 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) |
16 |
12 13 14 15
|
syl3anc |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜 ) ) → 𝑜 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) |
17 |
|
ineq1 |
⊢ ( 𝑛 = 𝑜 → ( 𝑛 ∩ 𝑠 ) = ( 𝑜 ∩ 𝑠 ) ) |
18 |
17
|
neeq1d |
⊢ ( 𝑛 = 𝑜 → ( ( 𝑛 ∩ 𝑠 ) ≠ ∅ ↔ ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) |
19 |
18
|
ralbidv |
⊢ ( 𝑛 = 𝑜 → ( ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ ↔ ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) |
20 |
19
|
rspcv |
⊢ ( 𝑜 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) → ( ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) |
21 |
16 20
|
syl |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜 ) ) → ( ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) |
22 |
21
|
expr |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑜 ∈ 𝐽 ) → ( 𝐴 ∈ 𝑜 → ( ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) |
23 |
22
|
com23 |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑜 ∈ 𝐽 ) → ( ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ → ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) |
24 |
23
|
ralrimdva |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) → ( ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ → ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) |
25 |
24
|
imdistanda |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ ) → ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) ) |
26 |
|
fclsopn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) ) |
27 |
25 26
|
sylibrd |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ ) → 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ) ) |
28 |
10 27
|
impbid |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∀ 𝑠 ∈ 𝐹 ( 𝑛 ∩ 𝑠 ) ≠ ∅ ) ) ) |