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Theorem filunirn 21896
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 6417 . . . . . 6 (fBas‘𝑦) ∈ V
21rabex 5007 . . . . 5 {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)} ∈ V
3 df-fil 21860 . . . . 5 Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)})
42, 3fnmpti 6229 . . . 4 Fil Fn V
5 fnunirn 6731 . . . 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 21895 . . . . . . 7 (𝐹 ∈ (Fil‘𝑥) → 𝐹 = 𝑥)
87fveq2d 6408 . . . . . 6 (𝐹 ∈ (Fil‘𝑥) → (Fil‘ 𝐹) = (Fil‘𝑥))
98eleq2d 2871 . . . . 5 (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥)))
109ibir 259 . . . 4 (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
1110rexlimivw 3217 . . 3 (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
126, 11sylbi 208 . 2 (𝐹 ran Fil → 𝐹 ∈ (Fil‘ 𝐹))
13 fvssunirn 6433 . . 3 (Fil‘ 𝐹) ⊆ ran Fil
1413sseli 3794 . 2 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ran Fil)
1512, 14impbii 200 1 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wcel 2156  wne 2978  wral 3096  wrex 3097  {crab 3100  Vcvv 3391  cin 3768  c0 4116  𝒫 cpw 4351   cuni 4630  ran crn 5312   Fn wfn 6092  cfv 6097  fBascfbas 19938  Filcfil 21859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-fv 6105  df-fbas 19947  df-fil 21860
This theorem is referenced by:  flimfil  21983  isfcls  22023
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