Theorem cfilss

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

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

Proof

Step	Hyp	Ref	Expression
1		simprl	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → 𝐺 ∈ ( Fil ‘ 𝑋 ) )
2		simprr	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → 𝐹 ⊆ 𝐺 )
3		iscfil	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ↔ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) )
4	3	simplbda	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) )
5	4	adantr	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) )
6		ssrexv	⊢ ( 𝐹 ⊆ 𝐺 → ( ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) → ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) )
7	6	ralimdv	⊢ ( 𝐹 ⊆ 𝐺 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) )
8	2 5 7	sylc	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) )
9		iscfil	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐺 ∈ ( CauFil ‘ 𝐷 ) ↔ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) )
10	9	ad2antrr	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → ( 𝐺 ∈ ( CauFil ‘ 𝐷 ) ↔ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) )
11	1 8 10	mpbir2and	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → 𝐺 ∈ ( CauFil ‘ 𝐷 ) )