Step |
Hyp |
Ref |
Expression |
1 |
|
metdscn.f |
|- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) ) |
2 |
|
metdscn.j |
|- J = ( MetOpen ` D ) |
3 |
|
metdscn2.k |
|- K = ( TopOpen ` CCfld ) |
4 |
|
eqid |
|- ( dist ` RR*s ) = ( dist ` RR*s ) |
5 |
4
|
xrsdsre |
|- ( ( dist ` RR*s ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) ) |
6 |
4
|
xrsxmet |
|- ( dist ` RR*s ) e. ( *Met ` RR* ) |
7 |
|
ressxr |
|- RR C_ RR* |
8 |
|
eqid |
|- ( ( dist ` RR*s ) |` ( RR X. RR ) ) = ( ( dist ` RR*s ) |` ( RR X. RR ) ) |
9 |
|
eqid |
|- ( MetOpen ` ( dist ` RR*s ) ) = ( MetOpen ` ( dist ` RR*s ) ) |
10 |
|
eqid |
|- ( MetOpen ` ( ( dist ` RR*s ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( dist ` RR*s ) |` ( RR X. RR ) ) ) |
11 |
8 9 10
|
metrest |
|- ( ( ( dist ` RR*s ) e. ( *Met ` RR* ) /\ RR C_ RR* ) -> ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) = ( MetOpen ` ( ( dist ` RR*s ) |` ( RR X. RR ) ) ) ) |
12 |
6 7 11
|
mp2an |
|- ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) = ( MetOpen ` ( ( dist ` RR*s ) |` ( RR X. RR ) ) ) |
13 |
5 12
|
tgioo |
|- ( topGen ` ran (,) ) = ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) |
14 |
3
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( K |`t RR ) |
15 |
13 14
|
eqtr3i |
|- ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) = ( K |`t RR ) |
16 |
15
|
oveq2i |
|- ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) ) = ( J Cn ( K |`t RR ) ) |
17 |
3
|
cnfldtop |
|- K e. Top |
18 |
|
cnrest2r |
|- ( K e. Top -> ( J Cn ( K |`t RR ) ) C_ ( J Cn K ) ) |
19 |
17 18
|
ax-mp |
|- ( J Cn ( K |`t RR ) ) C_ ( J Cn K ) |
20 |
16 19
|
eqsstri |
|- ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) ) C_ ( J Cn K ) |
21 |
|
metxmet |
|- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) ) |
22 |
1 2 4 9
|
metdscn |
|- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) ) |
23 |
21 22
|
sylan |
|- ( ( D e. ( Met ` X ) /\ S C_ X ) -> F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) ) |
24 |
23
|
3adant3 |
|- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) ) |
25 |
1
|
metdsre |
|- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F : X --> RR ) |
26 |
|
frn |
|- ( F : X --> RR -> ran F C_ RR ) |
27 |
9
|
mopntopon |
|- ( ( dist ` RR*s ) e. ( *Met ` RR* ) -> ( MetOpen ` ( dist ` RR*s ) ) e. ( TopOn ` RR* ) ) |
28 |
6 27
|
ax-mp |
|- ( MetOpen ` ( dist ` RR*s ) ) e. ( TopOn ` RR* ) |
29 |
|
cnrest2 |
|- ( ( ( MetOpen ` ( dist ` RR*s ) ) e. ( TopOn ` RR* ) /\ ran F C_ RR /\ RR C_ RR* ) -> ( F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) <-> F e. ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) ) ) ) |
30 |
28 7 29
|
mp3an13 |
|- ( ran F C_ RR -> ( F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) <-> F e. ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) ) ) ) |
31 |
25 26 30
|
3syl |
|- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> ( F e. ( J Cn ( MetOpen ` ( dist ` RR*s ) ) ) <-> F e. ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) ) ) ) |
32 |
24 31
|
mpbid |
|- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F e. ( J Cn ( ( MetOpen ` ( dist ` RR*s ) ) |`t RR ) ) ) |
33 |
20 32
|
sselid |
|- ( ( D e. ( Met ` X ) /\ S C_ X /\ S =/= (/) ) -> F e. ( J Cn K ) ) |