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Theorem filunirn 21906
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 6342 . . . . . 6 (fBas‘𝑦) ∈ V
21rabex 4946 . . . . 5 {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)} ∈ V
3 df-fil 21870 . . . . 5 Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)})
42, 3fnmpti 6162 . . . 4 Fil Fn V
5 fnunirn 6654 . . . 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 21905 . . . . . . 7 (𝐹 ∈ (Fil‘𝑥) → 𝐹 = 𝑥)
87fveq2d 6336 . . . . . 6 (𝐹 ∈ (Fil‘𝑥) → (Fil‘ 𝐹) = (Fil‘𝑥))
98eleq2d 2836 . . . . 5 (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥)))
109ibir 257 . . . 4 (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
1110rexlimivw 3177 . . 3 (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
126, 11sylbi 207 . 2 (𝐹 ran Fil → 𝐹 ∈ (Fil‘ 𝐹))
13 fvssunirn 6358 . . 3 (Fil‘ 𝐹) ⊆ ran Fil
1413sseli 3748 . 2 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ran Fil)
1512, 14impbii 199 1 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  cin 3722  c0 4063  𝒫 cpw 4297   cuni 4574  ran crn 5250   Fn wfn 6026  cfv 6031  fBascfbas 19949  Filcfil 21869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-fbas 19958  df-fil 21870
This theorem is referenced by:  flimfil  21993  isfcls  22033
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