Step |
Hyp |
Ref |
Expression |
1 |
|
setsms.x |
|- ( ph -> X = ( Base ` M ) ) |
2 |
|
setsms.d |
|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) ) |
3 |
|
setsms.k |
|- ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) |
4 |
|
setsms.m |
|- ( ph -> M e. V ) |
5 |
1 2 3 4
|
setsmstset |
|- ( ph -> ( MetOpen ` D ) = ( TopSet ` K ) ) |
6 |
|
df-mopn |
|- MetOpen = ( x e. U. ran *Met |-> ( topGen ` ran ( ball ` x ) ) ) |
7 |
6
|
dmmptss |
|- dom MetOpen C_ U. ran *Met |
8 |
7
|
sseli |
|- ( D e. dom MetOpen -> D e. U. ran *Met ) |
9 |
|
simpr |
|- ( ( ph /\ D e. U. ran *Met ) -> D e. U. ran *Met ) |
10 |
|
xmetunirn |
|- ( D e. U. ran *Met <-> D e. ( *Met ` dom dom D ) ) |
11 |
9 10
|
sylib |
|- ( ( ph /\ D e. U. ran *Met ) -> D e. ( *Met ` dom dom D ) ) |
12 |
|
eqid |
|- ( MetOpen ` D ) = ( MetOpen ` D ) |
13 |
12
|
mopnuni |
|- ( D e. ( *Met ` dom dom D ) -> dom dom D = U. ( MetOpen ` D ) ) |
14 |
11 13
|
syl |
|- ( ( ph /\ D e. U. ran *Met ) -> dom dom D = U. ( MetOpen ` D ) ) |
15 |
2
|
dmeqd |
|- ( ph -> dom D = dom ( ( dist ` M ) |` ( X X. X ) ) ) |
16 |
|
dmres |
|- dom ( ( dist ` M ) |` ( X X. X ) ) = ( ( X X. X ) i^i dom ( dist ` M ) ) |
17 |
15 16
|
eqtrdi |
|- ( ph -> dom D = ( ( X X. X ) i^i dom ( dist ` M ) ) ) |
18 |
|
inss1 |
|- ( ( X X. X ) i^i dom ( dist ` M ) ) C_ ( X X. X ) |
19 |
17 18
|
eqsstrdi |
|- ( ph -> dom D C_ ( X X. X ) ) |
20 |
|
dmss |
|- ( dom D C_ ( X X. X ) -> dom dom D C_ dom ( X X. X ) ) |
21 |
19 20
|
syl |
|- ( ph -> dom dom D C_ dom ( X X. X ) ) |
22 |
|
dmxpid |
|- dom ( X X. X ) = X |
23 |
21 22
|
sseqtrdi |
|- ( ph -> dom dom D C_ X ) |
24 |
23
|
adantr |
|- ( ( ph /\ D e. U. ran *Met ) -> dom dom D C_ X ) |
25 |
14 24
|
eqsstrrd |
|- ( ( ph /\ D e. U. ran *Met ) -> U. ( MetOpen ` D ) C_ X ) |
26 |
|
sspwuni |
|- ( ( MetOpen ` D ) C_ ~P X <-> U. ( MetOpen ` D ) C_ X ) |
27 |
25 26
|
sylibr |
|- ( ( ph /\ D e. U. ran *Met ) -> ( MetOpen ` D ) C_ ~P X ) |
28 |
27
|
ex |
|- ( ph -> ( D e. U. ran *Met -> ( MetOpen ` D ) C_ ~P X ) ) |
29 |
8 28
|
syl5 |
|- ( ph -> ( D e. dom MetOpen -> ( MetOpen ` D ) C_ ~P X ) ) |
30 |
|
ndmfv |
|- ( -. D e. dom MetOpen -> ( MetOpen ` D ) = (/) ) |
31 |
|
0ss |
|- (/) C_ ~P X |
32 |
30 31
|
eqsstrdi |
|- ( -. D e. dom MetOpen -> ( MetOpen ` D ) C_ ~P X ) |
33 |
29 32
|
pm2.61d1 |
|- ( ph -> ( MetOpen ` D ) C_ ~P X ) |
34 |
1 2 3
|
setsmsbas |
|- ( ph -> X = ( Base ` K ) ) |
35 |
34
|
pweqd |
|- ( ph -> ~P X = ~P ( Base ` K ) ) |
36 |
33 5 35
|
3sstr3d |
|- ( ph -> ( TopSet ` K ) C_ ~P ( Base ` K ) ) |
37 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
38 |
|
eqid |
|- ( TopSet ` K ) = ( TopSet ` K ) |
39 |
37 38
|
topnid |
|- ( ( TopSet ` K ) C_ ~P ( Base ` K ) -> ( TopSet ` K ) = ( TopOpen ` K ) ) |
40 |
36 39
|
syl |
|- ( ph -> ( TopSet ` K ) = ( TopOpen ` K ) ) |
41 |
5 40
|
eqtrd |
|- ( ph -> ( MetOpen ` D ) = ( TopOpen ` K ) ) |