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

Theorem ssfg 23929
Description: A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
ssfg (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))

Proof of Theorem ssfg
Dummy variables 𝑥 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fbelss 23890 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡𝐹) → 𝑡𝑋)
21ex 416 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡𝑋))
3 ssid 3958 . . . . 5 𝑡𝑡
4 sseq1 3961 . . . . . 6 (𝑥 = 𝑡 → (𝑥𝑡𝑡𝑡))
54rspcev 3581 . . . . 5 ((𝑡𝐹𝑡𝑡) → ∃𝑥𝐹 𝑥𝑡)
63, 5mpan2 701 . . . 4 (𝑡𝐹 → ∃𝑥𝐹 𝑥𝑡)
72, 6jca2 521 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹 → (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
8 elfg 23928 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
97, 8sylibrd 261 . 2 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡 ∈ (𝑋filGen𝐹)))
109ssrdv 3942 1 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2142  wrex 3086  wss 3904  cfv 6521  (class class class)co 7396  fBascfbas 21409  filGencfg 21410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-fbas 21418  df-fg 21419
This theorem is referenced by:  fgss2  23931  fgfil  23932  fgabs  23936  trfg  23948  isufil2  23965  ssufl  23975  ufileu  23976  filufint  23977  elfm2  24005  fmfnfmlem4  24014  fmfnfm  24015  fmco  24018  hausflim  24038  flimclslem  24041  flffbas  24052  fclsbas  24078  fclsfnflim  24084  flimfnfcls  24085  fclscmp  24087  isucn2  24335  cfilufg  24349  metust  24615  psmetutop  24624  fgcfil  25330  cmetss  25375  minveclem4a  25489  minveclem4  25491  fgmin  36727
  Copyright terms: Public domain W3C validator