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Theorem ssfg 21581
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 21542 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡𝐹) → 𝑡𝑋)
21ex 450 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡𝑋))
3 ssid 3608 . . . . . 6 𝑡𝑡
4 sseq1 3610 . . . . . . 7 (𝑥 = 𝑡 → (𝑥𝑡𝑡𝑡))
54rspcev 3300 . . . . . 6 ((𝑡𝐹𝑡𝑡) → ∃𝑥𝐹 𝑥𝑡)
63, 5mpan2 706 . . . . 5 (𝑡𝐹 → ∃𝑥𝐹 𝑥𝑡)
76a1i 11 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹 → ∃𝑥𝐹 𝑥𝑡))
82, 7jcad 555 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹 → (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
9 elfg 21580 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
108, 9sylibrd 249 . 2 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡 ∈ (𝑋filGen𝐹)))
1110ssrdv 3594 1 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1992  wrex 2913  wss 3560  cfv 5850  (class class class)co 6605  fBascfbas 19648  filGencfg 19649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-fbas 19657  df-fg 19658
This theorem is referenced by:  fgss2  21583  fgfil  21584  fgabs  21588  trfg  21600  isufil2  21617  ssufl  21627  ufileu  21628  filufint  21629  elfm2  21657  fmfnfmlem4  21666  fmfnfm  21667  fmco  21670  hausflim  21690  flimclslem  21693  flffbas  21704  fclsbas  21730  fclsfnflim  21736  flimfnfcls  21737  fclscmp  21739  isucn2  21988  cfilufg  22002  metust  22268  psmetutop  22277  fgcfil  22972  cmetss  23016  minveclem4a  23104  minveclem4  23106  fgmin  31999
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