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Theorem filnet 34226
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 2739 . 2 𝑛𝐹 ({𝑛} × 𝑛) = 𝑛𝐹 ({𝑛} × 𝑛)
2 eqid 2739 . 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 34225 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 399   = wceq 1542  wex 1786  wcel 2114  wrex 3055  wss 3853  {csn 4526   ciun 4891  {copab 5102   × cxp 5533  dom cdm 5535  ran crn 5536  wf 6345  cfv 6349  (class class class)co 7182  1st c1st 7724  DirRelcdir 17966  tailctail 17967  Filcfil 22608   FilMap cfm 22696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-1st 7726  df-2nd 7727  df-dir 17968  df-tail 17969  df-fbas 20226  df-fg 20227  df-fil 22609  df-fm 22701
This theorem is referenced by: (None)
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