Theorem filfbas 22060

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.

Step	Hyp	Ref	Expression
1		isfil 22059	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹)))
2	1	simplbi 493	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2968  ∀wral 3089   ∩ cin 3790  ∅c0 4140  𝒫 cpw 4378  ‘cfv 6135  fBascfbas 20130  Filcfil 22057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143  df-fil 22058

This theorem is referenced by:  0nelfil  22061  filsspw  22063  filelss  22064  filin  22066  filtop  22067  snfbas  22078  fgfil  22087  elfilss  22088  filfinnfr  22089  fgabs  22091  filconn  22095  fgtr  22102  trfg  22103  ufilb  22118  ufilmax  22119  isufil2  22120  ssufl  22130  ufileu  22131  filufint  22132  ufilen  22142  fmfg  22161  fmufil  22171  fmid  22172  fmco  22173  ufldom  22174  hausflim  22193  flimrest  22195  flimclslem  22196  flfnei  22203  isflf  22205  flfcnp  22216  fclsrest  22236  fclsfnflim  22239  flimfnfcls  22240  isfcf  22246  cnpfcfi  22252  cnpfcf  22253  cnextcn  22279  cfilufg  22505  neipcfilu  22508  cnextucn  22515  ucnextcn  22516  cfilresi  23501  cfilres  23502  cmetss  23522  relcmpcmet  23524  cfilucfil3  23526  minveclem4a  23636  filnetlem4  32964