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 |
|
fvex |
|- ( MetOpen ` D ) e. _V |
6 |
|
tsetid |
|- TopSet = Slot ( TopSet ` ndx ) |
7 |
6
|
setsid |
|- ( ( M e. V /\ ( MetOpen ` D ) e. _V ) -> ( MetOpen ` D ) = ( TopSet ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) ) |
8 |
4 5 7
|
sylancl |
|- ( ph -> ( MetOpen ` D ) = ( TopSet ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) ) |
9 |
3
|
fveq2d |
|- ( ph -> ( TopSet ` K ) = ( TopSet ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) ) |
10 |
8 9
|
eqtr4d |
|- ( ph -> ( MetOpen ` D ) = ( TopSet ` K ) ) |