Step |
Hyp |
Ref |
Expression |
1 |
|
opprval.1 |
|- B = ( Base ` R ) |
2 |
|
opprval.2 |
|- .x. = ( .r ` R ) |
3 |
|
opprval.3 |
|- O = ( oppR ` R ) |
4 |
|
id |
|- ( x = R -> x = R ) |
5 |
|
fveq2 |
|- ( x = R -> ( .r ` x ) = ( .r ` R ) ) |
6 |
5 2
|
eqtr4di |
|- ( x = R -> ( .r ` x ) = .x. ) |
7 |
6
|
tposeqd |
|- ( x = R -> tpos ( .r ` x ) = tpos .x. ) |
8 |
7
|
opeq2d |
|- ( x = R -> <. ( .r ` ndx ) , tpos ( .r ` x ) >. = <. ( .r ` ndx ) , tpos .x. >. ) |
9 |
4 8
|
oveq12d |
|- ( x = R -> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) |
10 |
|
df-oppr |
|- oppR = ( x e. _V |-> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) ) |
11 |
|
ovex |
|- ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V |
12 |
9 10 11
|
fvmpt |
|- ( R e. _V -> ( oppR ` R ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) |
13 |
|
fvprc |
|- ( -. R e. _V -> ( oppR ` R ) = (/) ) |
14 |
|
reldmsets |
|- Rel dom sSet |
15 |
14
|
ovprc1 |
|- ( -. R e. _V -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) = (/) ) |
16 |
13 15
|
eqtr4d |
|- ( -. R e. _V -> ( oppR ` R ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) |
17 |
12 16
|
pm2.61i |
|- ( oppR ` R ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) |
18 |
3 17
|
eqtri |
|- O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) |