Metamath Proof Explorer


Theorem fibas

Description: A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009) (Revised by Mario Carneiro, 24-Nov-2013)

Ref Expression
Assertion fibas ( fi ‘ 𝐴 ) ∈ TopBases

Proof

Step Hyp Ref Expression
1 fvex ( fi ‘ 𝐴 ) ∈ V
2 fiin ( ( 𝑥 ∈ ( fi ‘ 𝐴 ) ∧ 𝑦 ∈ ( fi ‘ 𝐴 ) ) → ( 𝑥𝑦 ) ∈ ( fi ‘ 𝐴 ) )
3 2 rgen2 𝑥 ∈ ( fi ‘ 𝐴 ) ∀ 𝑦 ∈ ( fi ‘ 𝐴 ) ( 𝑥𝑦 ) ∈ ( fi ‘ 𝐴 )
4 fiinbas ( ( ( fi ‘ 𝐴 ) ∈ V ∧ ∀ 𝑥 ∈ ( fi ‘ 𝐴 ) ∀ 𝑦 ∈ ( fi ‘ 𝐴 ) ( 𝑥𝑦 ) ∈ ( fi ‘ 𝐴 ) ) → ( fi ‘ 𝐴 ) ∈ TopBases )
5 1 3 4 mp2an ( fi ‘ 𝐴 ) ∈ TopBases