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Theorem filunirn 21858
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 6350 . . . . . 6 (fBas‘𝑦) ∈ V
21rabex 4952 . . . . 5 {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)} ∈ V
3 df-fil 21822 . . . . 5 Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧𝑤)})
42, 3fnmpti 6171 . . . 4 Fil Fn V
5 fnunirn 6662 . . . 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 21857 . . . . . . 7 (𝐹 ∈ (Fil‘𝑥) → 𝐹 = 𝑥)
87fveq2d 6344 . . . . . 6 (𝐹 ∈ (Fil‘𝑥) → (Fil‘ 𝐹) = (Fil‘𝑥))
98eleq2d 2813 . . . . 5 (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥)))
109ibir 257 . . . 4 (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
1110rexlimivw 3155 . . 3 (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘ 𝐹))
126, 11sylbi 207 . 2 (𝐹 ran Fil → 𝐹 ∈ (Fil‘ 𝐹))
13 fvssunirn 6366 . . 3 (Fil‘ 𝐹) ⊆ ran Fil
1413sseli 3728 . 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 2127  wne 2920  wral 3038  wrex 3039  {crab 3042  Vcvv 3328  cin 3702  c0 4046  𝒫 cpw 4290   cuni 4576  ran crn 5255   Fn wfn 6032  cfv 6037  fBascfbas 19907  Filcfil 21821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045  df-fbas 19916  df-fil 21822
This theorem is referenced by:  flimfil  21945  isfcls  21985
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