Step |
Hyp |
Ref |
Expression |
1 |
|
lhpset.b |
|- B = ( Base ` K ) |
2 |
|
lhpset.u |
|- .1. = ( 1. ` K ) |
3 |
|
lhpset.c |
|- C = ( |
4 |
|
lhpset.h |
|- H = ( LHyp ` K ) |
5 |
|
elex |
|- ( K e. A -> K e. _V ) |
6 |
|
fveq2 |
|- ( k = K -> ( Base ` k ) = ( Base ` K ) ) |
7 |
6 1
|
eqtr4di |
|- ( k = K -> ( Base ` k ) = B ) |
8 |
|
eqidd |
|- ( k = K -> w = w ) |
9 |
|
fveq2 |
|- ( k = K -> ( |
10 |
9 3
|
eqtr4di |
|- ( k = K -> ( |
11 |
|
fveq2 |
|- ( k = K -> ( 1. ` k ) = ( 1. ` K ) ) |
12 |
11 2
|
eqtr4di |
|- ( k = K -> ( 1. ` k ) = .1. ) |
13 |
8 10 12
|
breq123d |
|- ( k = K -> ( w ( w C .1. ) ) |
14 |
7 13
|
rabeqbidv |
|- ( k = K -> { w e. ( Base ` k ) | w ( |
15 |
|
df-lhyp |
|- LHyp = ( k e. _V |-> { w e. ( Base ` k ) | w ( |
16 |
1
|
fvexi |
|- B e. _V |
17 |
16
|
rabex |
|- { w e. B | w C .1. } e. _V |
18 |
14 15 17
|
fvmpt |
|- ( K e. _V -> ( LHyp ` K ) = { w e. B | w C .1. } ) |
19 |
4 18
|
syl5eq |
|- ( K e. _V -> H = { w e. B | w C .1. } ) |
20 |
5 19
|
syl |
|- ( K e. A -> H = { w e. B | w C .1. } ) |