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Theorem filunirn 22494
 Description: Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
filunirn (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))

Proof of Theorem filunirn
Dummy variables 𝑦 𝑤 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6658 . . . . . 6 (fBas‘𝑦) ∈ V
21rabex 5199 . . . . 5 {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)} ∈ V
3 df-fil 22458 . . . . 5 Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)})
42, 3fnmpti 6463 . . . 4 Fil Fn V
5 fnunirn 6990 . . . 4 (Fil Fn V → (𝐹 ran Fil ↔ ∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥)))
64, 5ax-mp 5 . . 3 (𝐹 ran Fil ↔ ∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥))
7 filunibas 22493 . . . . . . 7 (𝐹 ∈ (Fil‘𝑥) → 𝐹 = 𝑥)
87fveq2d 6649 . . . . . 6 (𝐹 ∈ (Fil‘𝑥) → (Fil‘ 𝐹) = (Fil‘𝑥))
98eleq2d 2875 . . . . 5 (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥)))
109ibir 271 . . . 4 (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
1110rexlimivw 3241 . . 3 (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
126, 11sylbi 220 . 2 (𝐹 ran Fil → 𝐹 ∈ (Fil‘ 𝐹))
13 fvssunirn 6674 . . 3 (Fil‘ 𝐹) ⊆ ran Fil
1413sseli 3911 . 2 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ran Fil)
1512, 14impbii 212 1 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∩ cin 3880  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800  ran crn 5520   Fn wfn 6319  ‘cfv 6324  fBascfbas 20082  Filcfil 22457 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 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-fbas 20091  df-fil 22458 This theorem is referenced by:  flimfil  22581  isfcls  22621
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