Theorem iscfil

Description: The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015)

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

Proof

Step	Hyp	Ref	Expression
1		cfilfval	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( CauFil ‘ 𝐷 ) = { 𝑓 ∈ ( Fil ‘ 𝑋 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } )
2	1	eleq2d	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ↔ 𝐹 ∈ { 𝑓 ∈ ( Fil ‘ 𝑋 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ) )
3		rexeq	⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ↔ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) )
4	3	ralbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) )
5	4	elrab	⊢ ( 𝐹 ∈ { 𝑓 ∈ ( Fil ‘ 𝑋 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ↔ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) )
6	2 5	bitrdi	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ↔ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) )