Step |
Hyp |
Ref |
Expression |
1 |
|
opnregcld.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
1
|
clscld |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ∈ ( Clsd ‘ 𝐽 ) ) |
3 |
|
eqcom |
⊢ ( ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ↔ 𝐴 = ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) ) |
4 |
3
|
biimpi |
⊢ ( ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 → 𝐴 = ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) ) |
5 |
|
fveq2 |
⊢ ( 𝑐 = ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) → ( ( int ‘ 𝐽 ) ‘ 𝑐 ) = ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) ) |
6 |
5
|
rspceeqv |
⊢ ( ( ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ∈ ( Clsd ‘ 𝐽 ) ∧ 𝐴 = ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) ) → ∃ 𝑐 ∈ ( Clsd ‘ 𝐽 ) 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
7 |
2 4 6
|
syl2an |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) ∧ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ) → ∃ 𝑐 ∈ ( Clsd ‘ 𝐽 ) 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
8 |
7
|
ex |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 → ∃ 𝑐 ∈ ( Clsd ‘ 𝐽 ) 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) |
9 |
|
cldrcl |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → 𝐽 ∈ Top ) |
10 |
1
|
cldss |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → 𝑐 ⊆ 𝑋 ) |
11 |
1
|
ntrss2 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ 𝑐 ) |
12 |
9 10 11
|
syl2anc |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ 𝑐 ) |
13 |
1
|
clsss2 |
⊢ ( ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) ∧ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ 𝑐 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ⊆ 𝑐 ) |
14 |
12 13
|
mpdan |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ⊆ 𝑐 ) |
15 |
1
|
ntrss |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ⊆ 𝑐 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
16 |
9 10 14 15
|
syl3anc |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
17 |
1
|
ntridm |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
18 |
9 10 17
|
syl2anc |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
19 |
1
|
ntrss3 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ 𝑋 ) |
20 |
9 10 19
|
syl2anc |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ 𝑋 ) |
21 |
1
|
clsss3 |
⊢ ( ( 𝐽 ∈ Top ∧ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ⊆ 𝑋 ) |
22 |
9 20 21
|
syl2anc |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ⊆ 𝑋 ) |
23 |
1
|
sscls |
⊢ ( ( 𝐽 ∈ Top ∧ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) |
24 |
9 20 23
|
syl2anc |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) |
25 |
1
|
ntrss |
⊢ ( ( 𝐽 ∈ Top ∧ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ⊆ 𝑋 ∧ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) → ( ( int ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ⊆ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) ) |
26 |
9 22 24 25
|
syl3anc |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ⊆ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) ) |
27 |
18 26
|
eqsstrrd |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) ) |
28 |
16 27
|
eqssd |
⊢ ( 𝑐 ∈ ( Clsd ‘ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
29 |
28
|
adantl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑐 ∈ ( Clsd ‘ 𝐽 ) ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
30 |
|
2fveq3 |
⊢ ( 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) ) |
31 |
|
id |
⊢ ( 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) → 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) |
32 |
30 31
|
eqeq12d |
⊢ ( 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) → ( ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ↔ ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) |
33 |
29 32
|
syl5ibrcom |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑐 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ) ) |
34 |
33
|
rexlimdva |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ∃ 𝑐 ∈ ( Clsd ‘ 𝐽 ) 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) → ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ) ) |
35 |
8 34
|
impbid |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ( ( int ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) = 𝐴 ↔ ∃ 𝑐 ∈ ( Clsd ‘ 𝐽 ) 𝐴 = ( ( int ‘ 𝐽 ) ‘ 𝑐 ) ) ) |