| Step |
Hyp |
Ref |
Expression |
| 1 |
|
p1val.b |
|- B = ( Base ` K ) |
| 2 |
|
p1val.u |
|- U = ( lub ` K ) |
| 3 |
|
p1val.t |
|- .1. = ( 1. ` K ) |
| 4 |
|
elex |
|- ( K e. V -> K e. _V ) |
| 5 |
|
fveq2 |
|- ( k = K -> ( lub ` k ) = ( lub ` K ) ) |
| 6 |
5 2
|
eqtr4di |
|- ( k = K -> ( lub ` k ) = U ) |
| 7 |
|
fveq2 |
|- ( k = K -> ( Base ` k ) = ( Base ` K ) ) |
| 8 |
7 1
|
eqtr4di |
|- ( k = K -> ( Base ` k ) = B ) |
| 9 |
6 8
|
fveq12d |
|- ( k = K -> ( ( lub ` k ) ` ( Base ` k ) ) = ( U ` B ) ) |
| 10 |
|
df-p1 |
|- 1. = ( k e. _V |-> ( ( lub ` k ) ` ( Base ` k ) ) ) |
| 11 |
|
fvex |
|- ( U ` B ) e. _V |
| 12 |
9 10 11
|
fvmpt |
|- ( K e. _V -> ( 1. ` K ) = ( U ` B ) ) |
| 13 |
3 12
|
eqtrid |
|- ( K e. _V -> .1. = ( U ` B ) ) |
| 14 |
4 13
|
syl |
|- ( K e. V -> .1. = ( U ` B ) ) |