Metamath Proof Explorer


Theorem eltg

Description: Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006) (Revised by Mario Carneiro, 10-Jan-2015)

Ref Expression
Assertion eltg ( 𝐵𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ 𝐴 ( 𝐵 ∩ 𝒫 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 tgval ( 𝐵𝑉 → ( topGen ‘ 𝐵 ) = { 𝑥𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) } )
2 1 eleq2d ( 𝐵𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ 𝐴 ∈ { 𝑥𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) } ) )
3 elex ( 𝐴 ∈ { 𝑥𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) } → 𝐴 ∈ V )
4 3 adantl ( ( 𝐵𝑉𝐴 ∈ { 𝑥𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) } ) → 𝐴 ∈ V )
5 inex1g ( 𝐵𝑉 → ( 𝐵 ∩ 𝒫 𝐴 ) ∈ V )
6 5 uniexd ( 𝐵𝑉 ( 𝐵 ∩ 𝒫 𝐴 ) ∈ V )
7 ssexg ( ( 𝐴 ( 𝐵 ∩ 𝒫 𝐴 ) ∧ ( 𝐵 ∩ 𝒫 𝐴 ) ∈ V ) → 𝐴 ∈ V )
8 6 7 sylan2 ( ( 𝐴 ( 𝐵 ∩ 𝒫 𝐴 ) ∧ 𝐵𝑉 ) → 𝐴 ∈ V )
9 8 ancoms ( ( 𝐵𝑉𝐴 ( 𝐵 ∩ 𝒫 𝐴 ) ) → 𝐴 ∈ V )
10 id ( 𝑥 = 𝐴𝑥 = 𝐴 )
11 pweq ( 𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴 )
12 11 ineq2d ( 𝑥 = 𝐴 → ( 𝐵 ∩ 𝒫 𝑥 ) = ( 𝐵 ∩ 𝒫 𝐴 ) )
13 12 unieqd ( 𝑥 = 𝐴 ( 𝐵 ∩ 𝒫 𝑥 ) = ( 𝐵 ∩ 𝒫 𝐴 ) )
14 10 13 sseq12d ( 𝑥 = 𝐴 → ( 𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) ↔ 𝐴 ( 𝐵 ∩ 𝒫 𝐴 ) ) )
15 14 elabg ( 𝐴 ∈ V → ( 𝐴 ∈ { 𝑥𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) } ↔ 𝐴 ( 𝐵 ∩ 𝒫 𝐴 ) ) )
16 4 9 15 pm5.21nd ( 𝐵𝑉 → ( 𝐴 ∈ { 𝑥𝑥 ( 𝐵 ∩ 𝒫 𝑥 ) } ↔ 𝐴 ( 𝐵 ∩ 𝒫 𝐴 ) ) )
17 2 16 bitrd ( 𝐵𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ 𝐴 ( 𝐵 ∩ 𝒫 𝐴 ) ) )