Step |
Hyp |
Ref |
Expression |
1 |
|
0ex |
|- (/) e. _V |
2 |
|
satefv |
|- ( ( (/) e. _V /\ U e. V ) -> ( (/) SatE U ) = ( ( ( (/) Sat ( _E i^i ( (/) X. (/) ) ) ) ` _om ) ` U ) ) |
3 |
1 2
|
mpan |
|- ( U e. V -> ( (/) SatE U ) = ( ( ( (/) Sat ( _E i^i ( (/) X. (/) ) ) ) ` _om ) ` U ) ) |
4 |
|
xp0 |
|- ( (/) X. (/) ) = (/) |
5 |
4
|
ineq2i |
|- ( _E i^i ( (/) X. (/) ) ) = ( _E i^i (/) ) |
6 |
|
in0 |
|- ( _E i^i (/) ) = (/) |
7 |
5 6
|
eqtri |
|- ( _E i^i ( (/) X. (/) ) ) = (/) |
8 |
7
|
oveq2i |
|- ( (/) Sat ( _E i^i ( (/) X. (/) ) ) ) = ( (/) Sat (/) ) |
9 |
8
|
fveq1i |
|- ( ( (/) Sat ( _E i^i ( (/) X. (/) ) ) ) ` _om ) = ( ( (/) Sat (/) ) ` _om ) |
10 |
9
|
fveq1i |
|- ( ( ( (/) Sat ( _E i^i ( (/) X. (/) ) ) ) ` _om ) ` U ) = ( ( ( (/) Sat (/) ) ` _om ) ` U ) |
11 |
3 10
|
eqtrdi |
|- ( U e. V -> ( (/) SatE U ) = ( ( ( (/) Sat (/) ) ` _om ) ` U ) ) |