Step |
Hyp |
Ref |
Expression |
1 |
|
2fveq3 |
|- ( w = W -> ( Fil ` ( Base ` w ) ) = ( Fil ` ( Base ` W ) ) ) |
2 |
|
2fveq3 |
|- ( w = W -> ( CauFilU ` ( UnifSt ` w ) ) = ( CauFilU ` ( UnifSt ` W ) ) ) |
3 |
2
|
eleq2d |
|- ( w = W -> ( c e. ( CauFilU ` ( UnifSt ` w ) ) <-> c e. ( CauFilU ` ( UnifSt ` W ) ) ) ) |
4 |
|
fveq2 |
|- ( w = W -> ( TopOpen ` w ) = ( TopOpen ` W ) ) |
5 |
4
|
oveq1d |
|- ( w = W -> ( ( TopOpen ` w ) fLim c ) = ( ( TopOpen ` W ) fLim c ) ) |
6 |
5
|
neeq1d |
|- ( w = W -> ( ( ( TopOpen ` w ) fLim c ) =/= (/) <-> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) |
7 |
3 6
|
imbi12d |
|- ( w = W -> ( ( c e. ( CauFilU ` ( UnifSt ` w ) ) -> ( ( TopOpen ` w ) fLim c ) =/= (/) ) <-> ( c e. ( CauFilU ` ( UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) ) |
8 |
1 7
|
raleqbidv |
|- ( w = W -> ( A. c e. ( Fil ` ( Base ` w ) ) ( c e. ( CauFilU ` ( UnifSt ` w ) ) -> ( ( TopOpen ` w ) fLim c ) =/= (/) ) <-> A. c e. ( Fil ` ( Base ` W ) ) ( c e. ( CauFilU ` ( UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) ) |
9 |
|
df-cusp |
|- CUnifSp = { w e. UnifSp | A. c e. ( Fil ` ( Base ` w ) ) ( c e. ( CauFilU ` ( UnifSt ` w ) ) -> ( ( TopOpen ` w ) fLim c ) =/= (/) ) } |
10 |
8 9
|
elrab2 |
|- ( W e. CUnifSp <-> ( W e. UnifSp /\ A. c e. ( Fil ` ( Base ` W ) ) ( c e. ( CauFilU ` ( UnifSt ` W ) ) -> ( ( TopOpen ` W ) fLim c ) =/= (/) ) ) ) |