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 |
|
ffn |
|- ( F : ( Base ` S ) --> ( Base ` T ) -> F Fn ( Base ` S ) ) |
5 |
|
fnima |
|- ( F Fn ( Base ` S ) -> ( F " ( Base ` S ) ) = ran F ) |
6 |
3 4 5
|
3syl |
|- ( F e. ( S LMHom T ) -> ( F " ( Base ` S ) ) = ran F ) |
7 |
|
lmhmlmod1 |
|- ( F e. ( S LMHom T ) -> S e. LMod ) |
8 |
|
eqid |
|- ( LSubSp ` S ) = ( LSubSp ` S ) |
9 |
1 8
|
lss1 |
|- ( S e. LMod -> ( Base ` S ) e. ( LSubSp ` S ) ) |
10 |
7 9
|
syl |
|- ( F e. ( S LMHom T ) -> ( Base ` S ) e. ( LSubSp ` S ) ) |
11 |
|
eqid |
|- ( LSubSp ` T ) = ( LSubSp ` T ) |
12 |
8 11
|
lmhmima |
|- ( ( F e. ( S LMHom T ) /\ ( Base ` S ) e. ( LSubSp ` S ) ) -> ( F " ( Base ` S ) ) e. ( LSubSp ` T ) ) |
13 |
10 12
|
mpdan |
|- ( F e. ( S LMHom T ) -> ( F " ( Base ` S ) ) e. ( LSubSp ` T ) ) |
14 |
6 13
|
eqeltrrd |
|- ( F e. ( S LMHom T ) -> ran F e. ( LSubSp ` T ) ) |