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Theorem filnet 33346
 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 2795 . 2 𝑛𝐹 ({𝑛} × 𝑛) = 𝑛𝐹 ({𝑛} × 𝑛)
2 eqid 2795 . 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 33345 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 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081  ∃wrex 3106   ⊆ wss 3863  {csn 4476  ∪ ciun 4829  {copab 5028   × cxp 5446  dom cdm 5448  ran crn 5449  ⟶wf 6226  ‘cfv 6230  (class class class)co 7021  1st c1st 7548  DirRelcdir 17672  tailctail 17673  Filcfil 22142   FilMap cfm 22230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-1st 7550  df-2nd 7551  df-dir 17674  df-tail 17675  df-fbas 20229  df-fg 20230  df-fil 22143  df-fm 22235 This theorem is referenced by: (None)
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