Theorem cfilfil

Description: A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015)

	Ref	Expression
Assertion	cfilfil	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		iscfil	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ↔ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) )
2	1	simprbda	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) )