MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fgfil Structured version   Visualization version   GIF version

Theorem fgfil 22203
Description: A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fgfil (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)

Proof of Theorem fgfil
Dummy variables 𝑥 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filfbas 22176 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 elfg 22199 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
31, 2syl 17 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
4 filss 22181 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑡𝑋𝑥𝑡)) → 𝑡𝐹)
543exp2 1335 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑡𝑋 → (𝑥𝑡𝑡𝐹))))
65com34 91 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑥𝑡 → (𝑡𝑋𝑡𝐹))))
76rexlimdv 3223 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
87impcomd 403 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ((𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡) → 𝑡𝐹))
93, 8sylbid 232 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡𝐹))
109ssrdv 3859 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) ⊆ 𝐹)
11 ssfg 22200 . . 3 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
121, 11syl 17 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
1310, 12eqssd 3870 1 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wrex 3084  wss 3824  cfv 6186  (class class class)co 6975  fBascfbas 20251  filGencfg 20252  Filcfil 22173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-fbas 20260  df-fg 20261  df-fil 22174
This theorem is referenced by:  elfilss  22204  fgtr  22218  fmid  22288  isfcf  22362  cnextcn  22395  filnetlem4  33283
  Copyright terms: Public domain W3C validator