Theorem cfilufbas

Description: A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017)

	Ref	Expression
Assertion	cfilufbas	⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFilu ‘ 𝑈 ) ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		iscfilu	⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝐹 ∈ ( CauFilu ‘ 𝑈 ) ↔ ( 𝐹 ∈ ( fBas ‘ 𝑋 ) ∧ ∀ 𝑣 ∈ 𝑈 ∃ 𝑎 ∈ 𝐹 ( 𝑎 × 𝑎 ) ⊆ 𝑣 ) ) )
2	1	simprbda	⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFilu ‘ 𝑈 ) ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) )