Theorem ssfg 21676

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 e. ( fBas ` X ) -> F C_ ( X filGen F ) )

Proof of Theorem ssfg

Dummy variables x t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fbelss 21637	. . . . 5 |- ( ( F e. ( fBas ` X ) /\ t e. F ) -> t C_ X )
2	1	ex 450	. . . 4 |- ( F e. ( fBas ` X ) -> ( t e. F -> t C_ X ) )
3		ssid 3624	. . . . . 6 |- t C_ t
4		sseq1 3626	. . . . . . 7 |- ( x = t -> ( x C_ t <-> t C_ t ) )
5	4	rspcev 3309	. . . . . 6 |- ( ( t e. F /\ t C_ t ) -> E. x e. F x C_ t )
6	3, 5	mpan2 707	. . . . 5 |- ( t e. F -> E. x e. F x C_ t )
7	6	a1i 11	. . . 4 |- ( F e. ( fBas ` X ) -> ( t e. F -> E. x e. F x C_ t ) )
8	2, 7	jcad 555	. . 3 |- ( F e. ( fBas ` X ) -> ( t e. F -> ( t C_ X /\ E. x e. F x C_ t ) ) )
9		elfg 21675	. . 3 |- ( F e. ( fBas ` X ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. x e. F x C_ t ) ) )
10	8, 9	sylibrd 249	. 2 |- ( F e. ( fBas ` X ) -> ( t e. F -> t e. ( X filGen F ) ) )
11	10	ssrdv 3609	1 |- ( F e. ( fBas ` X ) -> F C_ ( X filGen F ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   E.wrex 2913   C_ wss 3574   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   filGencfg 19735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744

This theorem is referenced by:  fgss2  21678  fgfil  21679  fgabs  21683  trfg  21695  isufil2  21712  ssufl  21722  ufileu  21723  filufint  21724  elfm2  21752  fmfnfmlem4  21761  fmfnfm  21762  fmco  21765  hausflim  21785  flimclslem  21788  flffbas  21799  fclsbas  21825  fclsfnflim  21831  flimfnfcls  21832  fclscmp  21834  isucn2  22083  cfilufg  22097  metust  22363  psmetutop  22372  fgcfil  23069  cmetss  23113  minveclem4a  23201  minveclem4  23203  fgmin  32365