Step |
Hyp |
Ref |
Expression |
1 |
|
setsms.x |
⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝑀 ) ) |
2 |
|
setsms.d |
⊢ ( 𝜑 → 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) ) |
3 |
|
setsms.k |
⊢ ( 𝜑 → 𝐾 = ( 𝑀 sSet 〈 ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) 〉 ) ) |
4 |
|
setsms.m |
⊢ ( 𝜑 → 𝑀 ∈ 𝑉 ) |
5 |
1 2 3 4
|
setsmstset |
⊢ ( 𝜑 → ( MetOpen ‘ 𝐷 ) = ( TopSet ‘ 𝐾 ) ) |
6 |
|
df-mopn |
⊢ MetOpen = ( 𝑥 ∈ ∪ ran ∞Met ↦ ( topGen ‘ ran ( ball ‘ 𝑥 ) ) ) |
7 |
6
|
dmmptss |
⊢ dom MetOpen ⊆ ∪ ran ∞Met |
8 |
7
|
sseli |
⊢ ( 𝐷 ∈ dom MetOpen → 𝐷 ∈ ∪ ran ∞Met ) |
9 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met ) → 𝐷 ∈ ∪ ran ∞Met ) |
10 |
|
xmetunirn |
⊢ ( 𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ ( ∞Met ‘ dom dom 𝐷 ) ) |
11 |
9 10
|
sylib |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met ) → 𝐷 ∈ ( ∞Met ‘ dom dom 𝐷 ) ) |
12 |
|
eqid |
⊢ ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ 𝐷 ) |
13 |
12
|
mopnuni |
⊢ ( 𝐷 ∈ ( ∞Met ‘ dom dom 𝐷 ) → dom dom 𝐷 = ∪ ( MetOpen ‘ 𝐷 ) ) |
14 |
11 13
|
syl |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met ) → dom dom 𝐷 = ∪ ( MetOpen ‘ 𝐷 ) ) |
15 |
2
|
dmeqd |
⊢ ( 𝜑 → dom 𝐷 = dom ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) ) |
16 |
|
dmres |
⊢ dom ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) = ( ( 𝑋 × 𝑋 ) ∩ dom ( dist ‘ 𝑀 ) ) |
17 |
15 16
|
eqtrdi |
⊢ ( 𝜑 → dom 𝐷 = ( ( 𝑋 × 𝑋 ) ∩ dom ( dist ‘ 𝑀 ) ) ) |
18 |
|
inss1 |
⊢ ( ( 𝑋 × 𝑋 ) ∩ dom ( dist ‘ 𝑀 ) ) ⊆ ( 𝑋 × 𝑋 ) |
19 |
17 18
|
eqsstrdi |
⊢ ( 𝜑 → dom 𝐷 ⊆ ( 𝑋 × 𝑋 ) ) |
20 |
|
dmss |
⊢ ( dom 𝐷 ⊆ ( 𝑋 × 𝑋 ) → dom dom 𝐷 ⊆ dom ( 𝑋 × 𝑋 ) ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → dom dom 𝐷 ⊆ dom ( 𝑋 × 𝑋 ) ) |
22 |
|
dmxpid |
⊢ dom ( 𝑋 × 𝑋 ) = 𝑋 |
23 |
21 22
|
sseqtrdi |
⊢ ( 𝜑 → dom dom 𝐷 ⊆ 𝑋 ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met ) → dom dom 𝐷 ⊆ 𝑋 ) |
25 |
14 24
|
eqsstrrd |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met ) → ∪ ( MetOpen ‘ 𝐷 ) ⊆ 𝑋 ) |
26 |
|
sspwuni |
⊢ ( ( MetOpen ‘ 𝐷 ) ⊆ 𝒫 𝑋 ↔ ∪ ( MetOpen ‘ 𝐷 ) ⊆ 𝑋 ) |
27 |
25 26
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met ) → ( MetOpen ‘ 𝐷 ) ⊆ 𝒫 𝑋 ) |
28 |
27
|
ex |
⊢ ( 𝜑 → ( 𝐷 ∈ ∪ ran ∞Met → ( MetOpen ‘ 𝐷 ) ⊆ 𝒫 𝑋 ) ) |
29 |
8 28
|
syl5 |
⊢ ( 𝜑 → ( 𝐷 ∈ dom MetOpen → ( MetOpen ‘ 𝐷 ) ⊆ 𝒫 𝑋 ) ) |
30 |
|
ndmfv |
⊢ ( ¬ 𝐷 ∈ dom MetOpen → ( MetOpen ‘ 𝐷 ) = ∅ ) |
31 |
|
0ss |
⊢ ∅ ⊆ 𝒫 𝑋 |
32 |
30 31
|
eqsstrdi |
⊢ ( ¬ 𝐷 ∈ dom MetOpen → ( MetOpen ‘ 𝐷 ) ⊆ 𝒫 𝑋 ) |
33 |
29 32
|
pm2.61d1 |
⊢ ( 𝜑 → ( MetOpen ‘ 𝐷 ) ⊆ 𝒫 𝑋 ) |
34 |
1 2 3
|
setsmsbas |
⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝐾 ) ) |
35 |
34
|
pweqd |
⊢ ( 𝜑 → 𝒫 𝑋 = 𝒫 ( Base ‘ 𝐾 ) ) |
36 |
33 5 35
|
3sstr3d |
⊢ ( 𝜑 → ( TopSet ‘ 𝐾 ) ⊆ 𝒫 ( Base ‘ 𝐾 ) ) |
37 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
38 |
|
eqid |
⊢ ( TopSet ‘ 𝐾 ) = ( TopSet ‘ 𝐾 ) |
39 |
37 38
|
topnid |
⊢ ( ( TopSet ‘ 𝐾 ) ⊆ 𝒫 ( Base ‘ 𝐾 ) → ( TopSet ‘ 𝐾 ) = ( TopOpen ‘ 𝐾 ) ) |
40 |
36 39
|
syl |
⊢ ( 𝜑 → ( TopSet ‘ 𝐾 ) = ( TopOpen ‘ 𝐾 ) ) |
41 |
5 40
|
eqtrd |
⊢ ( 𝜑 → ( MetOpen ‘ 𝐷 ) = ( TopOpen ‘ 𝐾 ) ) |