Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
2 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
3 |
1 2
|
islmim |
|- ( F e. ( R LMIso S ) <-> ( F e. ( R LMHom S ) /\ F : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) ) |
4 |
1 2
|
lmhmf1o |
|- ( F e. ( R LMHom S ) -> ( F : ( Base ` R ) -1-1-onto-> ( Base ` S ) <-> `' F e. ( S LMHom R ) ) ) |
5 |
4
|
pm5.32i |
|- ( ( F e. ( R LMHom S ) /\ F : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) <-> ( F e. ( R LMHom S ) /\ `' F e. ( S LMHom R ) ) ) |
6 |
3 5
|
bitri |
|- ( F e. ( R LMIso S ) <-> ( F e. ( R LMHom S ) /\ `' F e. ( S LMHom R ) ) ) |