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Theorem ssfg 23881
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 23842 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡𝐹) → 𝑡𝑋)
21ex 412 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡𝑋))
3 ssid 4005 . . . . 5 𝑡𝑡
4 sseq1 4008 . . . . . 6 (𝑥 = 𝑡 → (𝑥𝑡𝑡𝑡))
54rspcev 3621 . . . . 5 ((𝑡𝐹𝑡𝑡) → ∃𝑥𝐹 𝑥𝑡)
63, 5mpan2 691 . . . 4 (𝑡𝐹 → ∃𝑥𝐹 𝑥𝑡)
72, 6jca2 513 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹 → (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
8 elfg 23880 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
97, 8sylibrd 259 . 2 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡 ∈ (𝑋filGen𝐹)))
109ssrdv 3988 1 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  wrex 3069  wss 3950  cfv 6560  (class class class)co 7432  fBascfbas 21353  filGencfg 21354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-fg 21363
This theorem is referenced by:  fgss2  23883  fgfil  23884  fgabs  23888  trfg  23900  isufil2  23917  ssufl  23927  ufileu  23928  filufint  23929  elfm2  23957  fmfnfmlem4  23966  fmfnfm  23967  fmco  23970  hausflim  23990  flimclslem  23993  flffbas  24004  fclsbas  24030  fclsfnflim  24036  flimfnfcls  24037  fclscmp  24039  isucn2  24289  cfilufg  24303  metust  24572  psmetutop  24581  fgcfil  25306  cmetss  25351  minveclem4a  25465  minveclem4  25467  fgmin  36372
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