Step |
Hyp |
Ref |
Expression |
1 |
|
connsubclo.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
connsubclo.3 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑋 ) |
3 |
|
connsubclo.4 |
⊢ ( 𝜑 → ( 𝐽 ↾t 𝐴 ) ∈ Conn ) |
4 |
|
connsubclo.5 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐽 ) |
5 |
|
connsubclo.6 |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) ≠ ∅ ) |
6 |
|
connsubclo.7 |
⊢ ( 𝜑 → 𝐵 ∈ ( Clsd ‘ 𝐽 ) ) |
7 |
|
eqid |
⊢ ∪ ( 𝐽 ↾t 𝐴 ) = ∪ ( 𝐽 ↾t 𝐴 ) |
8 |
|
cldrcl |
⊢ ( 𝐵 ∈ ( Clsd ‘ 𝐽 ) → 𝐽 ∈ Top ) |
9 |
6 8
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
10 |
1
|
topopn |
⊢ ( 𝐽 ∈ Top → 𝑋 ∈ 𝐽 ) |
11 |
9 10
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ 𝐽 ) |
12 |
11 2
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
13 |
|
elrestr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝐵 ∈ 𝐽 ) → ( 𝐵 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ) |
14 |
9 12 4 13
|
syl3anc |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ) |
15 |
|
eqid |
⊢ ( 𝐵 ∩ 𝐴 ) = ( 𝐵 ∩ 𝐴 ) |
16 |
|
ineq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∩ 𝐴 ) = ( 𝐵 ∩ 𝐴 ) ) |
17 |
16
|
rspceeqv |
⊢ ( ( 𝐵 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝐵 ∩ 𝐴 ) = ( 𝐵 ∩ 𝐴 ) ) → ∃ 𝑥 ∈ ( Clsd ‘ 𝐽 ) ( 𝐵 ∩ 𝐴 ) = ( 𝑥 ∩ 𝐴 ) ) |
18 |
6 15 17
|
sylancl |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( Clsd ‘ 𝐽 ) ( 𝐵 ∩ 𝐴 ) = ( 𝑥 ∩ 𝐴 ) ) |
19 |
1
|
restcld |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → ( ( 𝐵 ∩ 𝐴 ) ∈ ( Clsd ‘ ( 𝐽 ↾t 𝐴 ) ) ↔ ∃ 𝑥 ∈ ( Clsd ‘ 𝐽 ) ( 𝐵 ∩ 𝐴 ) = ( 𝑥 ∩ 𝐴 ) ) ) |
20 |
9 2 19
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐵 ∩ 𝐴 ) ∈ ( Clsd ‘ ( 𝐽 ↾t 𝐴 ) ) ↔ ∃ 𝑥 ∈ ( Clsd ‘ 𝐽 ) ( 𝐵 ∩ 𝐴 ) = ( 𝑥 ∩ 𝐴 ) ) ) |
21 |
18 20
|
mpbird |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) ∈ ( Clsd ‘ ( 𝐽 ↾t 𝐴 ) ) ) |
22 |
7 3 14 5 21
|
connclo |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) = ∪ ( 𝐽 ↾t 𝐴 ) ) |
23 |
1
|
restuni |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → 𝐴 = ∪ ( 𝐽 ↾t 𝐴 ) ) |
24 |
9 2 23
|
syl2anc |
⊢ ( 𝜑 → 𝐴 = ∪ ( 𝐽 ↾t 𝐴 ) ) |
25 |
22 24
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) = 𝐴 ) |
26 |
|
sseqin2 |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐴 ) |
27 |
25 26
|
sylibr |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |