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Theorem ssfg 23896
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 23857 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡𝐹) → 𝑡𝑋)
21ex 412 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡𝑋))
3 ssid 4018 . . . . 5 𝑡𝑡
4 sseq1 4021 . . . . . 6 (𝑥 = 𝑡 → (𝑥𝑡𝑡𝑡))
54rspcev 3622 . . . . 5 ((𝑡𝐹𝑡𝑡) → ∃𝑥𝐹 𝑥𝑡)
63, 5mpan2 691 . . . 4 (𝑡𝐹 → ∃𝑥𝐹 𝑥𝑡)
72, 6jca2 513 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹 → (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
8 elfg 23895 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
97, 8sylibrd 259 . 2 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡 ∈ (𝑋filGen𝐹)))
109ssrdv 4001 1 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wrex 3068  wss 3963  cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380
This theorem is referenced by:  fgss2  23898  fgfil  23899  fgabs  23903  trfg  23915  isufil2  23932  ssufl  23942  ufileu  23943  filufint  23944  elfm2  23972  fmfnfmlem4  23981  fmfnfm  23982  fmco  23985  hausflim  24005  flimclslem  24008  flffbas  24019  fclsbas  24045  fclsfnflim  24051  flimfnfcls  24052  fclscmp  24054  isucn2  24304  cfilufg  24318  metust  24587  psmetutop  24596  fgcfil  25319  cmetss  25364  minveclem4a  25478  minveclem4  25480  fgmin  36353
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