Step |
Hyp |
Ref |
Expression |
1 |
|
tuslem.k |
|- K = ( toUnifSp ` U ) |
2 |
1
|
tusunif |
|- ( U e. ( UnifOn ` X ) -> U = ( UnifSet ` K ) ) |
3 |
|
ustuni |
|- ( U e. ( UnifOn ` X ) -> U. U = ( X X. X ) ) |
4 |
2
|
unieqd |
|- ( U e. ( UnifOn ` X ) -> U. U = U. ( UnifSet ` K ) ) |
5 |
1
|
tusbas |
|- ( U e. ( UnifOn ` X ) -> X = ( Base ` K ) ) |
6 |
5
|
sqxpeqd |
|- ( U e. ( UnifOn ` X ) -> ( X X. X ) = ( ( Base ` K ) X. ( Base ` K ) ) ) |
7 |
3 4 6
|
3eqtr3rd |
|- ( U e. ( UnifOn ` X ) -> ( ( Base ` K ) X. ( Base ` K ) ) = U. ( UnifSet ` K ) ) |
8 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
9 |
|
eqid |
|- ( UnifSet ` K ) = ( UnifSet ` K ) |
10 |
8 9
|
ussid |
|- ( ( ( Base ` K ) X. ( Base ` K ) ) = U. ( UnifSet ` K ) -> ( UnifSet ` K ) = ( UnifSt ` K ) ) |
11 |
7 10
|
syl |
|- ( U e. ( UnifOn ` X ) -> ( UnifSet ` K ) = ( UnifSt ` K ) ) |
12 |
2 11
|
eqtrd |
|- ( U e. ( UnifOn ` X ) -> U = ( UnifSt ` K ) ) |