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Theorem isfil 21561
 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 21560 . 2 Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
2 pweq 4133 . . . 4 (𝑧 = 𝑋 → 𝒫 𝑧 = 𝒫 𝑋)
32adantr 481 . . 3 ((𝑧 = 𝑋𝑓 = 𝐹) → 𝒫 𝑧 = 𝒫 𝑋)
4 ineq1 3785 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥))
54neeq1d 2849 . . . . 5 (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅))
6 eleq2 2687 . . . . 5 (𝑓 = 𝐹 → (𝑥𝑓𝑥𝐹))
75, 6imbi12d 334 . . . 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 3141 . 2 ((𝑧 = 𝑋𝑓 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓) ↔ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
10 fveq2 6148 . 2 (𝑧 = 𝑋 → (fBas‘𝑧) = (fBas‘𝑋))
11 fvex 6158 . 2 (fBas‘𝑧) ∈ V
12 elfvdm 6177 . 2 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
131, 9, 10, 11, 12elmptrab2 21542 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 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907   ∩ cin 3554  ∅c0 3891  𝒫 cpw 4130  dom cdm 5074  ‘cfv 5847  fBascfbas 19653  Filcfil 21559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-fil 21560 This theorem is referenced by:  filfbas  21562  filss  21567  isfil2  21570  ustfilxp  21926
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