Step |
Hyp |
Ref |
Expression |
1 |
|
releq |
|- ( r = R -> ( Rel r <-> Rel R ) ) |
2 |
|
coeq1 |
|- ( r = R -> ( r o. r ) = ( R o. r ) ) |
3 |
|
coeq2 |
|- ( r = R -> ( R o. r ) = ( R o. R ) ) |
4 |
2 3
|
eqtrd |
|- ( r = R -> ( r o. r ) = ( R o. R ) ) |
5 |
|
id |
|- ( r = R -> r = R ) |
6 |
4 5
|
sseq12d |
|- ( r = R -> ( ( r o. r ) C_ r <-> ( R o. R ) C_ R ) ) |
7 |
|
cnveq |
|- ( r = R -> `' r = `' R ) |
8 |
5 7
|
ineq12d |
|- ( r = R -> ( r i^i `' r ) = ( R i^i `' R ) ) |
9 |
|
unieq |
|- ( r = R -> U. r = U. R ) |
10 |
9
|
unieqd |
|- ( r = R -> U. U. r = U. U. R ) |
11 |
10
|
reseq2d |
|- ( r = R -> ( _I |` U. U. r ) = ( _I |` U. U. R ) ) |
12 |
8 11
|
eqeq12d |
|- ( r = R -> ( ( r i^i `' r ) = ( _I |` U. U. r ) <-> ( R i^i `' R ) = ( _I |` U. U. R ) ) ) |
13 |
1 6 12
|
3anbi123d |
|- ( r = R -> ( ( Rel r /\ ( r o. r ) C_ r /\ ( r i^i `' r ) = ( _I |` U. U. r ) ) <-> ( Rel R /\ ( R o. R ) C_ R /\ ( R i^i `' R ) = ( _I |` U. U. R ) ) ) ) |
14 |
|
df-ps |
|- PosetRel = { r | ( Rel r /\ ( r o. r ) C_ r /\ ( r i^i `' r ) = ( _I |` U. U. r ) ) } |
15 |
13 14
|
elab2g |
|- ( R e. A -> ( R e. PosetRel <-> ( Rel R /\ ( R o. R ) C_ R /\ ( R i^i `' R ) = ( _I |` U. U. R ) ) ) ) |