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Theorem filnet 36383
Description: A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Assertion
Ref Expression
filnet (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
Distinct variable groups:   𝑓,𝑑,𝐹   𝑋,𝑑,𝑓

Proof of Theorem filnet
Dummy variables 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . 2 𝑛𝐹 ({𝑛} × 𝑛) = 𝑛𝐹 ({𝑛} × 𝑛)
2 eqid 2737 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 𝑛𝐹 ({𝑛} × 𝑛) ∧ 𝑦 𝑛𝐹 ({𝑛} × 𝑛)) ∧ (1st𝑦) ⊆ (1st𝑥))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 𝑛𝐹 ({𝑛} × 𝑛) ∧ 𝑦 𝑛𝐹 ({𝑛} × 𝑛)) ∧ (1st𝑦) ⊆ (1st𝑥))}
31, 2filnetlem4 36382 1 (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  wss 3951  {csn 4626   ciun 4991  {copab 5205   × cxp 5683  dom cdm 5685  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  DirRelcdir 18639  tailctail 18640  Filcfil 23853   FilMap cfm 23941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-dir 18641  df-tail 18642  df-fbas 21361  df-fg 21362  df-fil 23854  df-fm 23946
This theorem is referenced by: (None)
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