Step |
Hyp |
Ref |
Expression |
1 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) → 𝐽 ∈ Top ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → 𝐽 ∈ Top ) |
3 |
|
simp2 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → 𝐴 ⊆ 𝑌 ) |
4 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) → 𝑌 = ∪ 𝐽 ) |
5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → 𝑌 = ∪ 𝐽 ) |
6 |
3 5
|
sseqtrd |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → 𝐴 ⊆ ∪ 𝐽 ) |
7 |
|
simp3 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → 𝑃 ∈ 𝑌 ) |
8 |
7 5
|
eleqtrd |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → 𝑃 ∈ ∪ 𝐽 ) |
9 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
10 |
9
|
neindisj2 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽 ) → ( 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ↔ ∀ 𝑣 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ( 𝑣 ∩ 𝐴 ) ≠ ∅ ) ) |
11 |
2 6 8 10
|
syl3anc |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → ( 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ↔ ∀ 𝑣 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ( 𝑣 ∩ 𝐴 ) ≠ ∅ ) ) |
12 |
|
simp1 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → 𝐽 ∈ ( TopOn ‘ 𝑌 ) ) |
13 |
7
|
snssd |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → { 𝑃 } ⊆ 𝑌 ) |
14 |
|
snnzg |
⊢ ( 𝑃 ∈ 𝑌 → { 𝑃 } ≠ ∅ ) |
15 |
14
|
3ad2ant3 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → { 𝑃 } ≠ ∅ ) |
16 |
|
neifil |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ { 𝑃 } ⊆ 𝑌 ∧ { 𝑃 } ≠ ∅ ) → ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∈ ( Fil ‘ 𝑌 ) ) |
17 |
12 13 15 16
|
syl3anc |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∈ ( Fil ‘ 𝑌 ) ) |
18 |
|
trfil2 |
⊢ ( ( ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ∈ ( Fil ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ) → ( ( ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ↾t 𝐴 ) ∈ ( Fil ‘ 𝐴 ) ↔ ∀ 𝑣 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ( 𝑣 ∩ 𝐴 ) ≠ ∅ ) ) |
19 |
17 3 18
|
syl2anc |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → ( ( ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ↾t 𝐴 ) ∈ ( Fil ‘ 𝐴 ) ↔ ∀ 𝑣 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ( 𝑣 ∩ 𝐴 ) ≠ ∅ ) ) |
20 |
11 19
|
bitr4d |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌 ) → ( 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ↔ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) ↾t 𝐴 ) ∈ ( Fil ‘ 𝐴 ) ) ) |