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Theorem filnet 32714
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 2771 . 2 𝑛𝐹 ({𝑛} × 𝑛) = 𝑛𝐹 ({𝑛} × 𝑛)
2 eqid 2771 . 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 32713 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 382   = wceq 1631  wex 1852  wcel 2145  wrex 3062  wss 3723  {csn 4316   ciun 4654  {copab 4846   × cxp 5247  dom cdm 5249  ran crn 5250  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  DirRelcdir 17436  tailctail 17437  Filcfil 21869   FilMap cfm 21957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-dir 17438  df-tail 17439  df-fbas 19958  df-fg 19959  df-fil 21870  df-fm 21962
This theorem is referenced by: (None)
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