Theorem ssfg 21897

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.

Step	Hyp	Ref	Expression
1		fbelss 21858	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡 ∈ 𝐹) → 𝑡 ⊆ 𝑋)
2	1	ex 449	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ⊆ 𝑋))
3		ssid 3765	. . . . . 6 ⊢ 𝑡 ⊆ 𝑡
4		sseq1 3767	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡))
5	4	rspcev 3449	. . . . . 6 ⊢ ((𝑡 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑡) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)
6	3, 5	mpan2 709	. . . . 5 ⊢ (𝑡 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)
7	6	a1i 11	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))
8	2, 7	jcad 556	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)))
9		elfg 21896	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)))
10	8, 9	sylibrd 249	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ∈ (𝑋filGen𝐹)))
11	10	ssrdv 3750	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2139  ∃wrex 3051   ⊆ wss 3715  ‘cfv 6049  (class class class)co 6814  fBascfbas 19956  filGencfg 19957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-fbas 19965  df-fg 19966

This theorem is referenced by:  fgss2  21899  fgfil  21900  fgabs  21904  trfg  21916  isufil2  21933  ssufl  21943  ufileu  21944  filufint  21945  elfm2  21973  fmfnfmlem4  21982  fmfnfm  21983  fmco  21986  hausflim  22006  flimclslem  22009  flffbas  22020  fclsbas  22046  fclsfnflim  22052  flimfnfcls  22053  fclscmp  22055  isucn2  22304  cfilufg  22318  metust  22584  psmetutop  22593  fgcfil  23289  cmetss  23333  minveclem4a  23421  minveclem4  23423  fgmin  32692