Step |
Hyp |
Ref |
Expression |
1 |
|
isfcls2 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ↔ ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) ) |
2 |
|
filn0 |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ≠ ∅ ) |
3 |
2
|
adantl |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → 𝐹 ≠ ∅ ) |
4 |
|
r19.2z |
⊢ ( ( 𝐹 ≠ ∅ ∧ ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) → ∃ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) |
5 |
4
|
ex |
⊢ ( 𝐹 ≠ ∅ → ( ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → ∃ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) ) |
6 |
3 5
|
syl |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → ∃ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) ) |
7 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
8 |
7
|
ad2antrr |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑠 ∈ 𝐹 ) → 𝐽 ∈ Top ) |
9 |
|
filelss |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → 𝑠 ⊆ 𝑋 ) |
10 |
9
|
adantll |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑠 ∈ 𝐹 ) → 𝑠 ⊆ 𝑋 ) |
11 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑠 ∈ 𝐹 ) → 𝑋 = ∪ 𝐽 ) |
13 |
10 12
|
sseqtrd |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑠 ∈ 𝐹 ) → 𝑠 ⊆ ∪ 𝐽 ) |
14 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
15 |
14
|
clsss3 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ⊆ ∪ 𝐽 ) |
16 |
8 13 15
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑠 ∈ 𝐹 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ⊆ ∪ 𝐽 ) |
17 |
16 12
|
sseqtrrd |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑠 ∈ 𝐹 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ⊆ 𝑋 ) |
18 |
17
|
sseld |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑠 ∈ 𝐹 ) → ( 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → 𝐴 ∈ 𝑋 ) ) |
19 |
18
|
rexlimdva |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( ∃ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → 𝐴 ∈ 𝑋 ) ) |
20 |
6 19
|
syld |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → 𝐴 ∈ 𝑋 ) ) |
21 |
20
|
pm4.71rd |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) ) ) |
22 |
7
|
ad3antrrr |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → 𝐽 ∈ Top ) |
23 |
13
|
adantlr |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → 𝑠 ⊆ ∪ 𝐽 ) |
24 |
|
simplr |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → 𝐴 ∈ 𝑋 ) |
25 |
11
|
ad3antrrr |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → 𝑋 = ∪ 𝐽 ) |
26 |
24 25
|
eleqtrd |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → 𝐴 ∈ ∪ 𝐽 ) |
27 |
14
|
elcls |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽 ∧ 𝐴 ∈ ∪ 𝐽 ) → ( 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) |
28 |
22 23 26 27
|
syl3anc |
⊢ ( ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑠 ∈ 𝐹 ) → ( 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) |
29 |
28
|
ralbidva |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) → ( ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ↔ ∀ 𝑠 ∈ 𝐹 ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) |
30 |
|
ralcom |
⊢ ( ∀ 𝑠 ∈ 𝐹 ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ↔ ∀ 𝑜 ∈ 𝐽 ∀ 𝑠 ∈ 𝐹 ( 𝐴 ∈ 𝑜 → ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) |
31 |
|
r19.21v |
⊢ ( ∀ 𝑠 ∈ 𝐹 ( 𝐴 ∈ 𝑜 → ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ↔ ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) |
32 |
31
|
ralbii |
⊢ ( ∀ 𝑜 ∈ 𝐽 ∀ 𝑠 ∈ 𝐹 ( 𝐴 ∈ 𝑜 → ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) |
33 |
30 32
|
bitri |
⊢ ( ∀ 𝑠 ∈ 𝐹 ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) |
34 |
29 33
|
bitrdi |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝑋 ) → ( ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ↔ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) |
35 |
34
|
pm5.32da |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) ) |
36 |
1 21 35
|
3bitrd |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) ) |