Step |
Hyp |
Ref |
Expression |
1 |
|
pltval.l |
|- .<_ = ( le ` K ) |
2 |
|
pltval.s |
|- .< = ( lt ` K ) |
3 |
|
elex |
|- ( K e. A -> K e. _V ) |
4 |
|
fveq2 |
|- ( p = K -> ( le ` p ) = ( le ` K ) ) |
5 |
4 1
|
eqtr4di |
|- ( p = K -> ( le ` p ) = .<_ ) |
6 |
5
|
difeq1d |
|- ( p = K -> ( ( le ` p ) \ _I ) = ( .<_ \ _I ) ) |
7 |
|
df-plt |
|- lt = ( p e. _V |-> ( ( le ` p ) \ _I ) ) |
8 |
1
|
fvexi |
|- .<_ e. _V |
9 |
8
|
difexi |
|- ( .<_ \ _I ) e. _V |
10 |
6 7 9
|
fvmpt |
|- ( K e. _V -> ( lt ` K ) = ( .<_ \ _I ) ) |
11 |
3 10
|
syl |
|- ( K e. A -> ( lt ` K ) = ( .<_ \ _I ) ) |
12 |
2 11
|
eqtrid |
|- ( K e. A -> .< = ( .<_ \ _I ) ) |