Step |
Hyp |
Ref |
Expression |
1 |
|
fiss |
|- ( ( J e. Top /\ A C_ J ) -> ( fi ` A ) C_ ( fi ` J ) ) |
2 |
|
fitop |
|- ( J e. Top -> ( fi ` J ) = J ) |
3 |
2
|
adantr |
|- ( ( J e. Top /\ A C_ J ) -> ( fi ` J ) = J ) |
4 |
1 3
|
sseqtrd |
|- ( ( J e. Top /\ A C_ J ) -> ( fi ` A ) C_ J ) |
5 |
|
tgss |
|- ( ( J e. Top /\ ( fi ` A ) C_ J ) -> ( topGen ` ( fi ` A ) ) C_ ( topGen ` J ) ) |
6 |
4 5
|
syldan |
|- ( ( J e. Top /\ A C_ J ) -> ( topGen ` ( fi ` A ) ) C_ ( topGen ` J ) ) |
7 |
|
tgtop |
|- ( J e. Top -> ( topGen ` J ) = J ) |
8 |
7
|
adantr |
|- ( ( J e. Top /\ A C_ J ) -> ( topGen ` J ) = J ) |
9 |
6 8
|
sseqtrd |
|- ( ( J e. Top /\ A C_ J ) -> ( topGen ` ( fi ` A ) ) C_ J ) |