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Theorem filfbas 21871
Description: A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
filfbas (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))

Proof of Theorem filfbas
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfil 21870 . 2 (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
21simplbi 479 1 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  wne 2942  wral 3060  cin 3720  c0 4061  𝒫 cpw 4295  cfv 6031  fBascfbas 19948  Filcfil 21868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  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-fv 6039  df-fil 21869
This theorem is referenced by:  0nelfil  21872  filsspw  21874  filelss  21875  filin  21877  filtop  21878  snfbas  21889  fgfil  21898  elfilss  21899  filfinnfr  21900  fgabs  21902  filconn  21906  fgtr  21913  trfg  21914  ufilb  21929  ufilmax  21930  isufil2  21931  ssufl  21941  ufileu  21942  filufint  21943  ufilen  21953  fmfg  21972  fmufil  21982  fmid  21983  fmco  21984  ufldom  21985  hausflim  22004  flimrest  22006  flimclslem  22007  flfnei  22014  isflf  22016  flfcnp  22027  fclsrest  22047  fclsfnflim  22050  flimfnfcls  22051  isfcf  22057  cnpfcfi  22063  cnpfcf  22064  cnextcn  22090  cfilufg  22316  neipcfilu  22319  cnextucn  22326  ucnextcn  22327  cfilresi  23311  cfilres  23312  cmetss  23331  relcmpcmet  23333  cfilucfil3  23335  minveclem4a  23419  filnetlem4  32707
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