Theorem metuel

Description: Elementhood in the uniform structure generated by a metric D (Contributed by Thierry Arnoux, 8-Dec-2017) (Revised by Thierry Arnoux, 11-Feb-2018)

	Ref	Expression
Assertion	metuel	⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( 𝑉 ∈ ( metUnif ‘ 𝐷 ) ↔ ( 𝑉 ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) 𝑤 ⊆ 𝑉 ) ) )

Proof

Step	Hyp	Ref	Expression
1		metuval	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( metUnif ‘ 𝐷 ) = ( ( 𝑋 × 𝑋 ) filGen ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) )
2	1	adantl	⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( metUnif ‘ 𝐷 ) = ( ( 𝑋 × 𝑋 ) filGen ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) )
3	2	eleq2d	⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( 𝑉 ∈ ( metUnif ‘ 𝐷 ) ↔ 𝑉 ∈ ( ( 𝑋 × 𝑋 ) filGen ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) ) )
4		oveq2	⊢ ( 𝑎 = 𝑒 → ( 0 [,) 𝑎 ) = ( 0 [,) 𝑒 ) )
5	4	imaeq2d	⊢ ( 𝑎 = 𝑒 → ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑒 ) ) )
6	5	cbvmptv	⊢ ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) = ( 𝑒 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑒 ) ) )
7	6	rneqi	⊢ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) = ran ( 𝑒 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑒 ) ) )
8	7	metustfbas	⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) )
9		elfg	⊢ ( ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∈ ( fBas ‘ ( 𝑋 × 𝑋 ) ) → ( 𝑉 ∈ ( ( 𝑋 × 𝑋 ) filGen ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) ↔ ( 𝑉 ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) 𝑤 ⊆ 𝑉 ) ) )
10	8 9	syl	⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( 𝑉 ∈ ( ( 𝑋 × 𝑋 ) filGen ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) ↔ ( 𝑉 ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) 𝑤 ⊆ 𝑉 ) ) )
11	3 10	bitrd	⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( 𝑉 ∈ ( metUnif ‘ 𝐷 ) ↔ ( 𝑉 ⊆ ( 𝑋 × 𝑋 ) ∧ ∃ 𝑤 ∈ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) 𝑤 ⊆ 𝑉 ) ) )