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

Theorem fgfil 22411
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 22384 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 elfg 22407 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
31, 2syl 17 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
4 filss 22389 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑡𝑋𝑥𝑡)) → 𝑡𝐹)
543exp2 1346 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑡𝑋 → (𝑥𝑡𝑡𝐹))))
65com34 91 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑥𝑡 → (𝑡𝑋𝑡𝐹))))
76rexlimdv 3280 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
87impcomd 412 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ((𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡) → 𝑡𝐹))
93, 8sylbid 241 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡𝐹))
109ssrdv 3970 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) ⊆ 𝐹)
11 ssfg 22408 . . 3 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
121, 11syl 17 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
1310, 12eqssd 3981 1 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  wss 3933  cfv 6348  (class class class)co 7145  fBascfbas 20461  filGencfg 20462  Filcfil 22381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fbas 20470  df-fg 20471  df-fil 22382
This theorem is referenced by:  elfilss  22412  fgtr  22426  fmid  22496  isfcf  22570  cnextcn  22603  filnetlem4  33626
  Copyright terms: Public domain W3C validator