Step |
Hyp |
Ref |
Expression |
1 |
|
is2ndc |
|- ( J e. 2ndc <-> E. x e. TopBases ( x ~<_ _om /\ ( topGen ` x ) = J ) ) |
2 |
|
simplr |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> x e. TopBases ) |
3 |
|
simpll |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> A e. V ) |
4 |
|
tgrest |
|- ( ( x e. TopBases /\ A e. V ) -> ( topGen ` ( x |`t A ) ) = ( ( topGen ` x ) |`t A ) ) |
5 |
2 3 4
|
syl2anc |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> ( topGen ` ( x |`t A ) ) = ( ( topGen ` x ) |`t A ) ) |
6 |
|
restbas |
|- ( x e. TopBases -> ( x |`t A ) e. TopBases ) |
7 |
6
|
ad2antlr |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> ( x |`t A ) e. TopBases ) |
8 |
|
restval |
|- ( ( x e. TopBases /\ A e. V ) -> ( x |`t A ) = ran ( y e. x |-> ( y i^i A ) ) ) |
9 |
2 3 8
|
syl2anc |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> ( x |`t A ) = ran ( y e. x |-> ( y i^i A ) ) ) |
10 |
|
1stcrestlem |
|- ( x ~<_ _om -> ran ( y e. x |-> ( y i^i A ) ) ~<_ _om ) |
11 |
10
|
adantl |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> ran ( y e. x |-> ( y i^i A ) ) ~<_ _om ) |
12 |
9 11
|
eqbrtrd |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> ( x |`t A ) ~<_ _om ) |
13 |
|
2ndci |
|- ( ( ( x |`t A ) e. TopBases /\ ( x |`t A ) ~<_ _om ) -> ( topGen ` ( x |`t A ) ) e. 2ndc ) |
14 |
7 12 13
|
syl2anc |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> ( topGen ` ( x |`t A ) ) e. 2ndc ) |
15 |
5 14
|
eqeltrrd |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> ( ( topGen ` x ) |`t A ) e. 2ndc ) |
16 |
|
oveq1 |
|- ( ( topGen ` x ) = J -> ( ( topGen ` x ) |`t A ) = ( J |`t A ) ) |
17 |
16
|
eleq1d |
|- ( ( topGen ` x ) = J -> ( ( ( topGen ` x ) |`t A ) e. 2ndc <-> ( J |`t A ) e. 2ndc ) ) |
18 |
15 17
|
syl5ibcom |
|- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ _om ) -> ( ( topGen ` x ) = J -> ( J |`t A ) e. 2ndc ) ) |
19 |
18
|
expimpd |
|- ( ( A e. V /\ x e. TopBases ) -> ( ( x ~<_ _om /\ ( topGen ` x ) = J ) -> ( J |`t A ) e. 2ndc ) ) |
20 |
19
|
rexlimdva |
|- ( A e. V -> ( E. x e. TopBases ( x ~<_ _om /\ ( topGen ` x ) = J ) -> ( J |`t A ) e. 2ndc ) ) |
21 |
1 20
|
syl5bi |
|- ( A e. V -> ( J e. 2ndc -> ( J |`t A ) e. 2ndc ) ) |
22 |
21
|
impcom |
|- ( ( J e. 2ndc /\ A e. V ) -> ( J |`t A ) e. 2ndc ) |