Step |
Hyp |
Ref |
Expression |
1 |
|
retopbas |
|- ran (,) e. TopBases |
2 |
|
bastg |
|- ( ran (,) e. TopBases -> ran (,) C_ ( topGen ` ran (,) ) ) |
3 |
1 2
|
ax-mp |
|- ran (,) C_ ( topGen ` ran (,) ) |
4 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
5 |
|
sssigagen |
|- ( ( topGen ` ran (,) ) e. Top -> ( topGen ` ran (,) ) C_ ( sigaGen ` ( topGen ` ran (,) ) ) ) |
6 |
4 5
|
ax-mp |
|- ( topGen ` ran (,) ) C_ ( sigaGen ` ( topGen ` ran (,) ) ) |
7 |
3 6
|
sstri |
|- ran (,) C_ ( sigaGen ` ( topGen ` ran (,) ) ) |
8 |
|
df-brsiga |
|- BrSiga = ( sigaGen ` ( topGen ` ran (,) ) ) |
9 |
7 8
|
sseqtrri |
|- ran (,) C_ BrSiga |
10 |
|
eqid |
|- vol = vol |
11 |
|
dmvlsiga |
|- dom vol e. ( sigAlgebra ` RR ) |
12 |
|
elrnsiga |
|- ( dom vol e. ( sigAlgebra ` RR ) -> dom vol e. U. ran sigAlgebra ) |
13 |
11 12
|
mp1i |
|- ( vol = vol -> dom vol e. U. ran sigAlgebra ) |
14 |
|
brsigarn |
|- BrSiga e. ( sigAlgebra ` RR ) |
15 |
|
elrnsiga |
|- ( BrSiga e. ( sigAlgebra ` RR ) -> BrSiga e. U. ran sigAlgebra ) |
16 |
14 15
|
mp1i |
|- ( vol = vol -> BrSiga e. U. ran sigAlgebra ) |
17 |
13 16
|
ismbfm |
|- ( vol = vol -> ( F e. ( dom vol MblFnM BrSiga ) <-> ( F e. ( U. BrSiga ^m U. dom vol ) /\ A. x e. BrSiga ( `' F " x ) e. dom vol ) ) ) |
18 |
10 17
|
ax-mp |
|- ( F e. ( dom vol MblFnM BrSiga ) <-> ( F e. ( U. BrSiga ^m U. dom vol ) /\ A. x e. BrSiga ( `' F " x ) e. dom vol ) ) |
19 |
18
|
simprbi |
|- ( F e. ( dom vol MblFnM BrSiga ) -> A. x e. BrSiga ( `' F " x ) e. dom vol ) |
20 |
|
ssralv |
|- ( ran (,) C_ BrSiga -> ( A. x e. BrSiga ( `' F " x ) e. dom vol -> A. x e. ran (,) ( `' F " x ) e. dom vol ) ) |
21 |
9 19 20
|
mpsyl |
|- ( F e. ( dom vol MblFnM BrSiga ) -> A. x e. ran (,) ( `' F " x ) e. dom vol ) |
22 |
18
|
simplbi |
|- ( F e. ( dom vol MblFnM BrSiga ) -> F e. ( U. BrSiga ^m U. dom vol ) ) |
23 |
|
elmapi |
|- ( F e. ( RR ^m RR ) -> F : RR --> RR ) |
24 |
|
unibrsiga |
|- U. BrSiga = RR |
25 |
|
unidmvol |
|- U. dom vol = RR |
26 |
24 25
|
oveq12i |
|- ( U. BrSiga ^m U. dom vol ) = ( RR ^m RR ) |
27 |
23 26
|
eleq2s |
|- ( F e. ( U. BrSiga ^m U. dom vol ) -> F : RR --> RR ) |
28 |
|
ismbf |
|- ( F : RR --> RR -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) ) |
29 |
22 27 28
|
3syl |
|- ( F e. ( dom vol MblFnM BrSiga ) -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) ) |
30 |
21 29
|
mpbird |
|- ( F e. ( dom vol MblFnM BrSiga ) -> F e. MblFn ) |