Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
2 |
|
eqid |
|- ( Base ` T ) = ( Base ` T ) |
3 |
1 2
|
lmhmf |
|- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) ) |
4 |
|
frel |
|- ( F : ( Base ` S ) --> ( Base ` T ) -> Rel F ) |
5 |
3 4
|
syl |
|- ( F e. ( S LMHom T ) -> Rel F ) |
6 |
|
dfrel2 |
|- ( Rel F <-> `' `' F = F ) |
7 |
5 6
|
sylib |
|- ( F e. ( S LMHom T ) -> `' `' F = F ) |
8 |
|
id |
|- ( F e. ( S LMHom T ) -> F e. ( S LMHom T ) ) |
9 |
7 8
|
eqeltrd |
|- ( F e. ( S LMHom T ) -> `' `' F e. ( S LMHom T ) ) |
10 |
9
|
anim1ci |
|- ( ( F e. ( S LMHom T ) /\ `' F e. ( T LMHom S ) ) -> ( `' F e. ( T LMHom S ) /\ `' `' F e. ( S LMHom T ) ) ) |
11 |
|
islmim2 |
|- ( F e. ( S LMIso T ) <-> ( F e. ( S LMHom T ) /\ `' F e. ( T LMHom S ) ) ) |
12 |
|
islmim2 |
|- ( `' F e. ( T LMIso S ) <-> ( `' F e. ( T LMHom S ) /\ `' `' F e. ( S LMHom T ) ) ) |
13 |
10 11 12
|
3imtr4i |
|- ( F e. ( S LMIso T ) -> `' F e. ( T LMIso S ) ) |