Theorem fgmin 32365

Description: Minimality property of a generated filter: every filter that contains B contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)

Assertion

Ref	Expression
fgmin	|- ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) -> ( B C_ F <-> ( X filGen B ) C_ F ) )

Proof of Theorem fgmin

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

Step	Hyp	Ref	Expression
1		elfg 21675	. . . . . . 7 |- ( B e. ( fBas ` X ) -> ( t e. ( X filGen B ) <-> ( t C_ X /\ E. x e. B x C_ t ) ) )
2	1	adantr 481	. . . . . 6 |- ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) -> ( t e. ( X filGen B ) <-> ( t C_ X /\ E. x e. B x C_ t ) ) )
3	2	adantr 481	. . . . 5 |- ( ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) /\ B C_ F ) -> ( t e. ( X filGen B ) <-> ( t C_ X /\ E. x e. B x C_ t ) ) )
4		ssrexv 3667	. . . . . . . . 9 |- ( B C_ F -> ( E. x e. B x C_ t -> E. x e. F x C_ t ) )
5	4	adantl 482	. . . . . . . 8 |- ( ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) /\ B C_ F ) -> ( E. x e. B x C_ t -> E. x e. F x C_ t ) )
6		filss 21657	. . . . . . . . . . . 12 |- ( ( F e. ( Fil ` X ) /\ ( x e. F /\ t C_ X /\ x C_ t ) ) -> t e. F )
7	6	3exp2 1285	. . . . . . . . . . 11 |- ( F e. ( Fil ` X ) -> ( x e. F -> ( t C_ X -> ( x C_ t -> t e. F ) ) ) )
8	7	com34 91	. . . . . . . . . 10 |- ( F e. ( Fil ` X ) -> ( x e. F -> ( x C_ t -> ( t C_ X -> t e. F ) ) ) )
9	8	rexlimdv 3030	. . . . . . . . 9 |- ( F e. ( Fil ` X ) -> ( E. x e. F x C_ t -> ( t C_ X -> t e. F ) ) )
10	9	ad2antlr 763	. . . . . . . 8 |- ( ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) /\ B C_ F ) -> ( E. x e. F x C_ t -> ( t C_ X -> t e. F ) ) )
11	5, 10	syld 47	. . . . . . 7 |- ( ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) /\ B C_ F ) -> ( E. x e. B x C_ t -> ( t C_ X -> t e. F ) ) )
12	11	com23 86	. . . . . 6 |- ( ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) /\ B C_ F ) -> ( t C_ X -> ( E. x e. B x C_ t -> t e. F ) ) )
13	12	impd 447	. . . . 5 |- ( ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) /\ B C_ F ) -> ( ( t C_ X /\ E. x e. B x C_ t ) -> t e. F ) )
14	3, 13	sylbid 230	. . . 4 |- ( ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) /\ B C_ F ) -> ( t e. ( X filGen B ) -> t e. F ) )
15	14	ssrdv 3609	. . 3 |- ( ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) /\ B C_ F ) -> ( X filGen B ) C_ F )
16	15	ex 450	. 2 |- ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) -> ( B C_ F -> ( X filGen B ) C_ F ) )
17		ssfg 21676	. . . 4 |- ( B e. ( fBas ` X ) -> B C_ ( X filGen B ) )
18		sstr2 3610	. . . 4 |- ( B C_ ( X filGen B ) -> ( ( X filGen B ) C_ F -> B C_ F ) )
19	17, 18	syl 17	. . 3 |- ( B e. ( fBas ` X ) -> ( ( X filGen B ) C_ F -> B C_ F ) )
20	19	adantr 481	. 2 |- ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) -> ( ( X filGen B ) C_ F -> B C_ F ) )
21	16, 20	impbid 202	1 |- ( ( B e. ( fBas ` X ) /\ F e. ( Fil ` X ) ) -> ( B C_ F <-> ( X filGen B ) C_ F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E.wrex 2913   C_ wss 3574   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   filGencfg 19735   Filcfil 21649

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  df-fil 21650

This theorem is referenced by: (None)