Metamath Proof Explorer

Theorem bastop1

Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " ( topGenB ) = J " to express " B is a basis for topology J " since we do not have a separate notation for this. Definition 15.35 of Schechter p. 428. (Contributed by NM, 2-Feb-2008) (Proof shortened by Mario Carneiro, 2-Sep-2015)

Ref Expression
Assertion bastop1 
|- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )

Proof

Step Hyp Ref Expression
1 tgss 
 |-  ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ ( topGen ` J ) )
2 tgtop 
 |-  ( J e. Top -> ( topGen ` J ) = J )
3 2 adantr 
 |-  ( ( J e. Top /\ B C_ J ) -> ( topGen ` J ) = J )
4 1 3 sseqtrd 
 |-  ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ J )
5 eqss 
 |-  ( ( topGen ` B ) = J <-> ( ( topGen ` B ) C_ J /\ J C_ ( topGen ` B ) ) )
6 5 baib 
 |-  ( ( topGen ` B ) C_ J -> ( ( topGen ` B ) = J <-> J C_ ( topGen ` B ) ) )
7 4 6 syl 
 |-  ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> J C_ ( topGen ` B ) ) )
8 dfss3 
 |-  ( J C_ ( topGen ` B ) <-> A. x e. J x e. ( topGen ` B ) )
9 7 8 bitrdi 
 |-  ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J x e. ( topGen ` B ) ) )
10 ssexg 
 |-  ( ( B C_ J /\ J e. Top ) -> B e. _V )
11 10 ancoms 
 |-  ( ( J e. Top /\ B C_ J ) -> B e. _V )
12 eltg3 
 |-  ( B e. _V -> ( x e. ( topGen ` B ) <-> E. y ( y C_ B /\ x = U. y ) ) )
13 11 12 syl 
 |-  ( ( J e. Top /\ B C_ J ) -> ( x e. ( topGen ` B ) <-> E. y ( y C_ B /\ x = U. y ) ) )
14 13 ralbidv 
 |-  ( ( J e. Top /\ B C_ J ) -> ( A. x e. J x e. ( topGen ` B ) <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )
15 9 14 bitrd 
 |-  ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )