Metamath Proof Explorer


Theorem tg1

Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006)

Ref Expression
Assertion tg1 ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → 𝐴 𝐵 )

Proof

Step Hyp Ref Expression
1 elfvdm ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → 𝐵 ∈ dom topGen )
2 eltg2 ( 𝐵 ∈ dom topGen → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ ( 𝐴 𝐵 ∧ ∀ 𝑥𝐴𝑦𝐵 ( 𝑥𝑦𝑦𝐴 ) ) ) )
3 2 simprbda ( ( 𝐵 ∈ dom topGen ∧ 𝐴 ∈ ( topGen ‘ 𝐵 ) ) → 𝐴 𝐵 )
4 1 3 mpancom ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → 𝐴 𝐵 )