Metamath Proof Explorer


Definition df-cfil

Description: Define the set of Cauchy filters on a given extended metric space. A Cauchy filter is a filter on the set such that for every 0 < x there is an element of the filter whose metric diameter is less than x . (Contributed by Mario Carneiro, 13-Oct-2015)

Ref Expression
Assertion df-cfil CauFil = ( 𝑑 ran ∞Met ↦ { 𝑓 ∈ ( Fil ‘ dom dom 𝑑 ) ∣ ∀ 𝑥 ∈ ℝ+𝑦𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } )

Detailed syntax breakdown

Step Hyp Ref Expression
0 ccfil CauFil
1 vd 𝑑
2 cxmet ∞Met
3 2 crn ran ∞Met
4 3 cuni ran ∞Met
5 vf 𝑓
6 cfil Fil
7 1 cv 𝑑
8 7 cdm dom 𝑑
9 8 cdm dom dom 𝑑
10 9 6 cfv ( Fil ‘ dom dom 𝑑 )
11 vx 𝑥
12 crp +
13 vy 𝑦
14 5 cv 𝑓
15 13 cv 𝑦
16 15 15 cxp ( 𝑦 × 𝑦 )
17 7 16 cima ( 𝑑 “ ( 𝑦 × 𝑦 ) )
18 cc0 0
19 cico [,)
20 11 cv 𝑥
21 18 20 19 co ( 0 [,) 𝑥 )
22 17 21 wss ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 )
23 22 13 14 wrex 𝑦𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 )
24 23 11 12 wral 𝑥 ∈ ℝ+𝑦𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 )
25 24 5 10 crab { 𝑓 ∈ ( Fil ‘ dom dom 𝑑 ) ∣ ∀ 𝑥 ∈ ℝ+𝑦𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) }
26 1 4 25 cmpt ( 𝑑 ran ∞Met ↦ { 𝑓 ∈ ( Fil ‘ dom dom 𝑑 ) ∣ ∀ 𝑥 ∈ ℝ+𝑦𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } )
27 0 26 wceq CauFil = ( 𝑑 ran ∞Met ↦ { 𝑓 ∈ ( Fil ‘ dom dom 𝑑 ) ∣ ∀ 𝑥 ∈ ℝ+𝑦𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } )