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Theorem isfil 21885
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 21884 . 2 Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
2 pweq 4365 . . . 4 (𝑧 = 𝑋 → 𝒫 𝑧 = 𝒫 𝑋)
32adantr 468 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → 𝒫 𝑧 = 𝒫 𝑋)
4 ineq1 4017 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
54neeq1d 3048 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
6 eleq2 2885 . . . . 5 (𝑓 = 𝐹 → (𝑥𝑓𝑥𝐹))
75, 6imbi12d 335 . . . 4 (𝑓 = 𝐹 → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
87adantl 469 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
93, 8raleqbidv 3352 . 2 ((𝑧 = 𝑋𝑓 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
10 fveq2 6418 . 2 (𝑧 = 𝑋 → (fBas‘𝑧) = (fBas‘𝑋))
11 fvex 6431 . 2 (fBas‘𝑧) ∈ V
12 elfvdm 6450 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
131, 9, 10, 11, 12elmptrab2 21866 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 197  wa 384   = wceq 1637  wcel 2157  wne 2989  wral 3107  cin 3779  c0 4127  𝒫 cpw 4362  dom cdm 5324  cfv 6111  fBascfbas 19962  Filcfil 21883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fv 6119  df-fil 21884
This theorem is referenced by:  filfbas  21886  filss  21891  isfil2  21894  ustfilxp  22250
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