Theorem fgss 21677

Description: A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fgss	|- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( X filGen F ) C_ ( X filGen G ) )

Proof of Theorem fgss

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

Step	Hyp	Ref	Expression
1		ssrexv 3667	. . . . 5 |- ( F C_ G -> ( E. x e. F x C_ t -> E. x e. G x C_ t ) )
2	1	anim2d 589	. . . 4 |- ( F C_ G -> ( ( t C_ X /\ E. x e. F x C_ t ) -> ( t C_ X /\ E. x e. G x C_ t ) ) )
3	2	3ad2ant3 1084	. . 3 |- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( ( t C_ X /\ E. x e. F x C_ t ) -> ( t C_ X /\ E. x e. G x C_ t ) ) )
4		elfg 21675	. . . 4 |- ( F e. ( fBas ` X ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. x e. F x C_ t ) ) )
5	4	3ad2ant1 1082	. . 3 |- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. x e. F x C_ t ) ) )
6		elfg 21675	. . . 4 |- ( G e. ( fBas ` X ) -> ( t e. ( X filGen G ) <-> ( t C_ X /\ E. x e. G x C_ t ) ) )
7	6	3ad2ant2 1083	. . 3 |- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( t e. ( X filGen G ) <-> ( t C_ X /\ E. x e. G x C_ t ) ) )
8	3, 5, 7	3imtr4d 283	. 2 |- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( t e. ( X filGen F ) -> t e. ( X filGen G ) ) )
9	8	ssrdv 3609	1 |- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( X filGen F ) C_ ( X filGen G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fg 19744

This theorem is referenced by:  fgabs  21683  fgtr  21694  fmss  21750