Step |
Hyp |
Ref |
Expression |
1 |
|
dibval.b |
|- B = ( Base ` K ) |
2 |
|
dibval.h |
|- H = ( LHyp ` K ) |
3 |
|
elex |
|- ( K e. V -> K e. _V ) |
4 |
|
fveq2 |
|- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) ) |
5 |
4 2
|
eqtr4di |
|- ( k = K -> ( LHyp ` k ) = H ) |
6 |
|
fveq2 |
|- ( k = K -> ( DIsoA ` k ) = ( DIsoA ` K ) ) |
7 |
6
|
fveq1d |
|- ( k = K -> ( ( DIsoA ` k ) ` w ) = ( ( DIsoA ` K ) ` w ) ) |
8 |
7
|
dmeqd |
|- ( k = K -> dom ( ( DIsoA ` k ) ` w ) = dom ( ( DIsoA ` K ) ` w ) ) |
9 |
7
|
fveq1d |
|- ( k = K -> ( ( ( DIsoA ` k ) ` w ) ` x ) = ( ( ( DIsoA ` K ) ` w ) ` x ) ) |
10 |
|
fveq2 |
|- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) ) |
11 |
10
|
fveq1d |
|- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) ) |
12 |
|
fveq2 |
|- ( k = K -> ( Base ` k ) = ( Base ` K ) ) |
13 |
12 1
|
eqtr4di |
|- ( k = K -> ( Base ` k ) = B ) |
14 |
13
|
reseq2d |
|- ( k = K -> ( _I |` ( Base ` k ) ) = ( _I |` B ) ) |
15 |
11 14
|
mpteq12dv |
|- ( k = K -> ( f e. ( ( LTrn ` k ) ` w ) |-> ( _I |` ( Base ` k ) ) ) = ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) ) |
16 |
15
|
sneqd |
|- ( k = K -> { ( f e. ( ( LTrn ` k ) ` w ) |-> ( _I |` ( Base ` k ) ) ) } = { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) |
17 |
9 16
|
xpeq12d |
|- ( k = K -> ( ( ( ( DIsoA ` k ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` k ) ` w ) |-> ( _I |` ( Base ` k ) ) ) } ) = ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) |
18 |
8 17
|
mpteq12dv |
|- ( k = K -> ( x e. dom ( ( DIsoA ` k ) ` w ) |-> ( ( ( ( DIsoA ` k ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` k ) ` w ) |-> ( _I |` ( Base ` k ) ) ) } ) ) = ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) |
19 |
5 18
|
mpteq12dv |
|- ( k = K -> ( w e. ( LHyp ` k ) |-> ( x e. dom ( ( DIsoA ` k ) ` w ) |-> ( ( ( ( DIsoA ` k ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` k ) ` w ) |-> ( _I |` ( Base ` k ) ) ) } ) ) ) = ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) ) |
20 |
|
df-dib |
|- DIsoB = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. dom ( ( DIsoA ` k ) ` w ) |-> ( ( ( ( DIsoA ` k ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` k ) ` w ) |-> ( _I |` ( Base ` k ) ) ) } ) ) ) ) |
21 |
19 20 2
|
mptfvmpt |
|- ( K e. _V -> ( DIsoB ` K ) = ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) ) |
22 |
3 21
|
syl |
|- ( K e. V -> ( DIsoB ` K ) = ( w e. H |-> ( x e. dom ( ( DIsoA ` K ) ` w ) |-> ( ( ( ( DIsoA ` K ) ` w ) ` x ) X. { ( f e. ( ( LTrn ` K ) ` w ) |-> ( _I |` B ) ) } ) ) ) ) |