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Theorem isfil 22560
 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 22559 . 2 Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
2 pweq 4513 . . . 4 (𝑧 = 𝑋 → 𝒫 𝑧 = 𝒫 𝑋)
32adantr 484 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → 𝒫 𝑧 = 𝒫 𝑋)
4 ineq1 4111 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
54neeq1d 3010 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
6 eleq2 2840 . . . . 5 (𝑓 = 𝐹 → (𝑥𝑓𝑥𝐹))
75, 6imbi12d 348 . . . 4 (𝑓 = 𝐹 → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
87adantl 485 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
93, 8raleqbidv 3319 . 2 ((𝑧 = 𝑋𝑓 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
10 fveq2 6663 . 2 (𝑧 = 𝑋 → (fBas‘𝑧) = (fBas‘𝑋))
11 fvex 6676 . 2 (fBas‘𝑧) ∈ V
12 elfvdm 6695 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
131, 9, 10, 11, 12elmptrab2 22541 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070   ∩ cin 3859  ∅c0 4227  𝒫 cpw 4497  dom cdm 5528  ‘cfv 6340  fBascfbas 20167  Filcfil 22558 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fv 6348  df-fil 22559 This theorem is referenced by:  filfbas  22561  filss  22566  isfil2  22569  ustfilxp  22926
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