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Theorem ssfg 21486
 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 21447 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡𝐹) → 𝑡𝑋)
21ex 449 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡𝑋))
3 ssid 3587 . . . . . 6 𝑡𝑡
4 sseq1 3589 . . . . . . 7 (𝑥 = 𝑡 → (𝑥𝑡𝑡𝑡))
54rspcev 3282 . . . . . 6 ((𝑡𝐹𝑡𝑡) → ∃𝑥𝐹 𝑥𝑡)
63, 5mpan2 703 . . . . 5 (𝑡𝐹 → ∃𝑥𝐹 𝑥𝑡)
76a1i 11 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹 → ∃𝑥𝐹 𝑥𝑡))
82, 7jcad 554 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹 → (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
9 elfg 21485 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
108, 9sylibrd 248 . 2 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡 ∈ (𝑋filGen𝐹)))
1110ssrdv 3574 1 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  fBascfbas 19555  filGencfg 19556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-fg 19565 This theorem is referenced by:  fgss2  21488  fgfil  21489  fgabs  21493  trfg  21505  isufil2  21522  ssufl  21532  ufileu  21533  filufint  21534  elfm2  21562  fmfnfmlem4  21571  fmfnfm  21572  fmco  21575  hausflim  21595  flimclslem  21598  flffbas  21609  fclsbas  21635  fclsfnflim  21641  flimfnfcls  21642  fclscmp  21644  isucn2  21893  cfilufg  21907  metust  22173  psmetutop  22182  fgcfil  22877  cmetss  22921  minveclem4a  23009  minveclem4  23011  fgmin  31535
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