Step |
Hyp |
Ref |
Expression |
1 |
|
opnregcld.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
1
|
ntropn |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∈ 𝐽 ) |
3 |
|
eqcom |
⊢ ( ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ↔ 𝐴 = ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ) |
4 |
3
|
biimpi |
⊢ ( ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 → 𝐴 = ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ) |
5 |
|
fveq2 |
⊢ ( 𝑜 = ( ( int ‘ 𝐽 ) ‘ 𝐴 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) = ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ) |
6 |
5
|
rspceeqv |
⊢ ( ( ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ∈ 𝐽 ∧ 𝐴 = ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) ) → ∃ 𝑜 ∈ 𝐽 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
7 |
2 4 6
|
syl2an |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) ∧ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ) → ∃ 𝑜 ∈ 𝐽 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
8 |
7
|
ex |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 → ∃ 𝑜 ∈ 𝐽 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) |
9 |
|
simpl |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → 𝐽 ∈ Top ) |
10 |
1
|
eltopss |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → 𝑜 ⊆ 𝑋 ) |
11 |
1
|
clsss3 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ⊆ 𝑋 ) |
12 |
10 11
|
syldan |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ⊆ 𝑋 ) |
13 |
1
|
ntrss2 |
⊢ ( ( 𝐽 ∈ Top ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
14 |
12 13
|
syldan |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
15 |
1
|
clsss |
⊢ ( ( 𝐽 ∈ Top ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ⊆ 𝑋 ∧ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) |
16 |
9 12 14 15
|
syl3anc |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) |
17 |
1
|
clsidm |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
18 |
10 17
|
syldan |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
19 |
16 18
|
sseqtrd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
20 |
1
|
ntrss3 |
⊢ ( ( 𝐽 ∈ Top ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ⊆ 𝑋 ) |
21 |
12 20
|
syldan |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ⊆ 𝑋 ) |
22 |
|
simpr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → 𝑜 ∈ 𝐽 ) |
23 |
1
|
sscls |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋 ) → 𝑜 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
24 |
10 23
|
syldan |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → 𝑜 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
25 |
1
|
ssntr |
⊢ ( ( ( 𝐽 ∈ Top ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ⊆ 𝑋 ) ∧ ( 𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) → 𝑜 ⊆ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) |
26 |
9 12 22 24 25
|
syl22anc |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → 𝑜 ⊆ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) |
27 |
1
|
clsss |
⊢ ( ( 𝐽 ∈ Top ∧ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ⊆ 𝑋 ∧ 𝑜 ⊆ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) ) |
28 |
9 21 26 27
|
syl3anc |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) ) |
29 |
19 28
|
eqssd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
30 |
29
|
adantlr |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑜 ∈ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
31 |
|
2fveq3 |
⊢ ( 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) ) |
32 |
|
id |
⊢ ( 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) → 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) |
33 |
31 32
|
eqeq12d |
⊢ ( 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) → ( ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ↔ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) |
34 |
30 33
|
syl5ibrcom |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑜 ∈ 𝐽 ) → ( 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ) ) |
35 |
34
|
rexlimdva |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ∃ 𝑜 ∈ 𝐽 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ) ) |
36 |
8 35
|
impbid |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ↔ ∃ 𝑜 ∈ 𝐽 𝐴 = ( ( cls ‘ 𝐽 ) ‘ 𝑜 ) ) ) |