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Theorem isfil 21871
Description: The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
isfil (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem isfil
Dummy variables 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fil 21870 . 2 Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
2 pweq 4300 . . . 4 (𝑧 = 𝑋 → 𝒫 𝑧 = 𝒫 𝑋)
32adantr 466 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → 𝒫 𝑧 = 𝒫 𝑋)
4 ineq1 3958 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
54neeq1d 3002 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
6 eleq2 2839 . . . . 5 (𝑓 = 𝐹 → (𝑥𝑓𝑥𝐹))
75, 6imbi12d 333 . . . 4 (𝑓 = 𝐹 → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
87adantl 467 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
93, 8raleqbidv 3301 . 2 ((𝑧 = 𝑋𝑓 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
10 fveq2 6332 . 2 (𝑧 = 𝑋 → (fBas‘𝑧) = (fBas‘𝑋))
11 fvex 6342 . 2 (fBas‘𝑧) ∈ V
12 elfvdm 6361 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
131, 9, 10, 11, 12elmptrab2 21852 1 (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  cin 3722  c0 4063  𝒫 cpw 4297  dom cdm 5249  cfv 6031  fBascfbas 19949  Filcfil 21869
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-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
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-ral 3066  df-rex 3067  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-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-fv 6039  df-fil 21870
This theorem is referenced by:  filfbas  21872  filss  21877  isfil2  21880  ustfilxp  22236
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