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Theorem filunirn 22210
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 6510 . . . . . 6 (fBas‘𝑦) ∈ V
21rabex 5088 . . . . 5 {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)} ∈ V
3 df-fil 22174 . . . . 5 Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)})
42, 3fnmpti 6319 . . . 4 Fil Fn V
5 fnunirn 6836 . . . 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 22209 . . . . . . 7 (𝐹 ∈ (Fil‘𝑥) → 𝐹 = 𝑥)
87fveq2d 6501 . . . . . 6 (𝐹 ∈ (Fil‘𝑥) → (Fil‘ 𝐹) = (Fil‘𝑥))
98eleq2d 2846 . . . . 5 (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥)))
109ibir 260 . . . 4 (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
1110rexlimivw 3222 . . 3 (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
126, 11sylbi 209 . 2 (𝐹 ran Fil → 𝐹 ∈ (Fil‘ 𝐹))
13 fvssunirn 6526 . . 3 (Fil‘ 𝐹) ⊆ ran Fil
1413sseli 3849 . 2 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ran Fil)
1512, 14impbii 201 1 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2051  wne 2962  wral 3083  wrex 3084  {crab 3087  Vcvv 3410  cin 3823  c0 4173  𝒫 cpw 4417   cuni 4709  ran crn 5405   Fn wfn 6181  cfv 6186  fBascfbas 20251  Filcfil 22173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-fv 6194  df-fbas 20260  df-fil 22174
This theorem is referenced by:  flimfil  22297  isfcls  22337
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