Step |
Hyp |
Ref |
Expression |
1 |
|
inss1 |
⊢ ( 𝐹 ∩ 𝐺 ) ⊆ 𝐹 |
2 |
|
filsspw |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝐹 ⊆ 𝒫 𝑋 ) |
3 |
2
|
adantr |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → 𝐹 ⊆ 𝒫 𝑋 ) |
4 |
1 3
|
sstrid |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐹 ∩ 𝐺 ) ⊆ 𝒫 𝑋 ) |
5 |
|
0nelfil |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ¬ ∅ ∈ 𝐹 ) |
6 |
5
|
adantr |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ¬ ∅ ∈ 𝐹 ) |
7 |
|
elinel1 |
⊢ ( ∅ ∈ ( 𝐹 ∩ 𝐺 ) → ∅ ∈ 𝐹 ) |
8 |
6 7
|
nsyl |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ¬ ∅ ∈ ( 𝐹 ∩ 𝐺 ) ) |
9 |
|
filtop |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝑋 ∈ 𝐹 ) |
10 |
9
|
adantr |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → 𝑋 ∈ 𝐹 ) |
11 |
|
filtop |
⊢ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) → 𝑋 ∈ 𝐺 ) |
12 |
11
|
adantl |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → 𝑋 ∈ 𝐺 ) |
13 |
10 12
|
elind |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → 𝑋 ∈ ( 𝐹 ∩ 𝐺 ) ) |
14 |
4 8 13
|
3jca |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ( ( 𝐹 ∩ 𝐺 ) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑋 ∈ ( 𝐹 ∩ 𝐺 ) ) ) |
15 |
|
simpll |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
16 |
|
simpr2 |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ) |
17 |
|
elinel1 |
⊢ ( 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) → 𝑦 ∈ 𝐹 ) |
18 |
16 17
|
syl |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑦 ∈ 𝐹 ) |
19 |
|
simpr1 |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ∈ 𝒫 𝑋 ) |
20 |
19
|
elpwid |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ⊆ 𝑋 ) |
21 |
|
simpr3 |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑦 ⊆ 𝑥 ) |
22 |
|
filss |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ∈ 𝐹 ) |
23 |
15 18 20 21 22
|
syl13anc |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ∈ 𝐹 ) |
24 |
|
simplr |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝐺 ∈ ( Fil ‘ 𝑋 ) ) |
25 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) → 𝑦 ∈ 𝐺 ) |
26 |
16 25
|
syl |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑦 ∈ 𝐺 ) |
27 |
|
filss |
⊢ ( ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑦 ∈ 𝐺 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ∈ 𝐺 ) |
28 |
24 26 20 21 27
|
syl13anc |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ∈ 𝐺 ) |
29 |
23 28
|
elind |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ) |
30 |
29
|
3exp2 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝑥 ∈ 𝒫 𝑋 → ( 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) → ( 𝑦 ⊆ 𝑥 → 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ) ) ) ) |
31 |
30
|
imp |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑥 ∈ 𝒫 𝑋 ) → ( 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) → ( 𝑦 ⊆ 𝑥 → 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ) ) ) |
32 |
31
|
rexlimdv |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ 𝑥 ∈ 𝒫 𝑋 ) → ( ∃ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) 𝑦 ⊆ 𝑥 → 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ) ) |
33 |
32
|
ralrimiva |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ∀ 𝑥 ∈ 𝒫 𝑋 ( ∃ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) 𝑦 ⊆ 𝑥 → 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ) ) |
34 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
35 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) → 𝑥 ∈ 𝐹 ) |
36 |
35 17
|
anim12i |
⊢ ( ( 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ) → ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) |
37 |
|
filin |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝐹 ) |
38 |
37
|
3expb |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝐹 ) |
39 |
34 36 38
|
syl2an |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ) ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝐹 ) |
40 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → 𝐺 ∈ ( Fil ‘ 𝑋 ) ) |
41 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) → 𝑥 ∈ 𝐺 ) |
42 |
41 25
|
anim12i |
⊢ ( ( 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ) → ( 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺 ) ) |
43 |
|
filin |
⊢ ( ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺 ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝐺 ) |
44 |
43
|
3expb |
⊢ ( ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺 ) ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝐺 ) |
45 |
40 42 44
|
syl2an |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ) ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝐺 ) |
46 |
39 45
|
elind |
⊢ ( ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) ∧ ( 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ) ) → ( 𝑥 ∩ 𝑦 ) ∈ ( 𝐹 ∩ 𝐺 ) ) |
47 |
46
|
ralrimivva |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ∀ 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ∀ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ( 𝑥 ∩ 𝑦 ) ∈ ( 𝐹 ∩ 𝐺 ) ) |
48 |
|
isfil2 |
⊢ ( ( 𝐹 ∩ 𝐺 ) ∈ ( Fil ‘ 𝑋 ) ↔ ( ( ( 𝐹 ∩ 𝐺 ) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ( 𝐹 ∩ 𝐺 ) ∧ 𝑋 ∈ ( 𝐹 ∩ 𝐺 ) ) ∧ ∀ 𝑥 ∈ 𝒫 𝑋 ( ∃ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) 𝑦 ⊆ 𝑥 → 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ) ∧ ∀ 𝑥 ∈ ( 𝐹 ∩ 𝐺 ) ∀ 𝑦 ∈ ( 𝐹 ∩ 𝐺 ) ( 𝑥 ∩ 𝑦 ) ∈ ( 𝐹 ∩ 𝐺 ) ) ) |
49 |
14 33 47 48
|
syl3anbrc |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝐹 ∩ 𝐺 ) ∈ ( Fil ‘ 𝑋 ) ) |