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Theorem isfil 22390
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 22389 . 2 Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
2 pweq 4545 . . . 4 (𝑧 = 𝑋 → 𝒫 𝑧 = 𝒫 𝑋)
32adantr 481 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → 𝒫 𝑧 = 𝒫 𝑋)
4 ineq1 4185 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
54neeq1d 3080 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
6 eleq2 2906 . . . . 5 (𝑓 = 𝐹 → (𝑥𝑓𝑥𝐹))
75, 6imbi12d 346 . . . 4 (𝑓 = 𝐹 → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
87adantl 482 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
93, 8raleqbidv 3407 . 2 ((𝑧 = 𝑋𝑓 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
10 fveq2 6669 . 2 (𝑧 = 𝑋 → (fBas‘𝑧) = (fBas‘𝑋))
11 fvex 6682 . 2 (fBas‘𝑧) ∈ V
12 elfvdm 6701 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
131, 9, 10, 11, 12elmptrab2 22371 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 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  cin 3939  c0 4295  𝒫 cpw 4542  dom cdm 5554  cfv 6354  fBascfbas 20468  Filcfil 22388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fv 6362  df-fil 22389
This theorem is referenced by:  filfbas  22391  filss  22396  isfil2  22399  ustfilxp  22755
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