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Theorem ssfg 22475
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 22436 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡𝐹) → 𝑡𝑋)
21ex 416 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡𝑋))
3 ssid 3964 . . . . 5 𝑡𝑡
4 sseq1 3967 . . . . . 6 (𝑥 = 𝑡 → (𝑥𝑡𝑡𝑡))
54rspcev 3598 . . . . 5 ((𝑡𝐹𝑡𝑡) → ∃𝑥𝐹 𝑥𝑡)
63, 5mpan2 690 . . . 4 (𝑡𝐹 → ∃𝑥𝐹 𝑥𝑡)
72, 6jca2 517 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹 → (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
8 elfg 22474 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
97, 8sylibrd 262 . 2 (𝐹 ∈ (fBas‘𝑋) → (𝑡𝐹𝑡 ∈ (𝑋filGen𝐹)))
109ssrdv 3948 1 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  wrex 3131  wss 3908  cfv 6334  (class class class)co 7140  fBascfbas 20077  filGencfg 20078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-fbas 20086  df-fg 20087
This theorem is referenced by:  fgss2  22477  fgfil  22478  fgabs  22482  trfg  22494  isufil2  22511  ssufl  22521  ufileu  22522  filufint  22523  elfm2  22551  fmfnfmlem4  22560  fmfnfm  22561  fmco  22564  hausflim  22584  flimclslem  22587  flffbas  22598  fclsbas  22624  fclsfnflim  22630  flimfnfcls  22631  fclscmp  22633  isucn2  22883  cfilufg  22897  metust  23163  psmetutop  23172  fgcfil  23873  cmetss  23918  minveclem4a  24032  minveclem4  24034  fgmin  33792
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