Theorem fgmin 15558

Description: Minimality property of a generated filter: every filter that contains B contains its generated filter.

Hypotheses

Ref	Expression
fgmin.1	|- X = U.B
fgmin.2	|- Y = U.F

Assertion

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

Proof of Theorem fgmin

Step	Hyp	Ref	Expression
1		fgmin.1	. . . . . . . 8 |- X = U.B
2	1	elfg 10284	. . . . . . 7 |- (B e. fBas -> (t e. (filGen` B) <-> (t C_ X /\ E.x e. B x C_ t)))
3	2	3ad2ant1 897	. . . . . 6 |- ((B e. fBas /\ F e. Fil /\ X = Y) -> (t e. (filGen` B) <-> (t C_ X /\ E.x e. B x C_ t)))
4	3	adantr 425	. . . . 5 |- (((B e. fBas /\ F e. Fil /\ X = Y) /\ B C_ F) -> (t e. (filGen` B) <-> (t C_ X /\ E.x e. B x C_ t)))
5		sseq2 2639	. . . . . . . . 9 |- (X = Y -> (t C_ X <-> t C_ Y))
6	5	3ad2ant3 899	. . . . . . . 8 |- ((B e. fBas /\ F e. Fil /\ X = Y) -> (t C_ X <-> t C_ Y))
7	6	adantr 425	. . . . . . 7 |- (((B e. fBas /\ F e. Fil /\ X = Y) /\ B C_ F) -> (t C_ X <-> t C_ Y))
8		ssrexv 2673	. . . . . . . . . . 11 |- (B C_ F -> (E.x e. B x C_ t -> E.x e. F x C_ t))
9	8	adantl 424	. . . . . . . . . 10 |- (((B e. fBas /\ F e. Fil) /\ B C_ F) -> (E.x e. B x C_ t -> E.x e. F x C_ t))
10		fgmin.2	. . . . . . . . . . . . . . 15 |- Y = U.F
11	10	fillsb 10270	. . . . . . . . . . . . . 14 |- (F e. Fil -> ((x e. F /\ t C_ Y /\ x C_ t) -> t e. F))
12	11	3expd 1085	. . . . . . . . . . . . 13 |- (F e. Fil -> (x e. F -> (t C_ Y -> (x C_ t -> t e. F))))
13	12	com34 40	. . . . . . . . . . . 12 |- (F e. Fil -> (x e. F -> (x C_ t -> (t C_ Y -> t e. F))))
14	13	r19.23adv 2215	. . . . . . . . . . 11 |- (F e. Fil -> (E.x e. F x C_ t -> (t C_ Y -> t e. F)))
15	14	ad2antlr 441	. . . . . . . . . 10 |- (((B e. fBas /\ F e. Fil) /\ B C_ F) -> (E.x e. F x C_ t -> (t C_ Y -> t e. F)))
16	9, 15	syld 30	. . . . . . . . 9 |- (((B e. fBas /\ F e. Fil) /\ B C_ F) -> (E.x e. B x C_ t -> (t C_ Y -> t e. F)))
17	16	com23 36	. . . . . . . 8 |- (((B e. fBas /\ F e. Fil) /\ B C_ F) -> (t C_ Y -> (E.x e. B x C_ t -> t e. F)))
18	17	3adantl3 1034	. . . . . . 7 |- (((B e. fBas /\ F e. Fil /\ X = Y) /\ B C_ F) -> (t C_ Y -> (E.x e. B x C_ t -> t e. F)))
19	7, 18	sylbid 220	. . . . . 6 |- (((B e. fBas /\ F e. Fil /\ X = Y) /\ B C_ F) -> (t C_ X -> (E.x e. B x C_ t -> t e. F)))
20	19	imp3a 388	. . . . 5 |- (((B e. fBas /\ F e. Fil /\ X = Y) /\ B C_ F) -> ((t C_ X /\ E.x e. B x C_ t) -> t e. F))
21	4, 20	sylbid 220	. . . 4 |- (((B e. fBas /\ F e. Fil /\ X = Y) /\ B C_ F) -> (t e. (filGen` B) -> t e. F))
22	21	ssrdv 2622	. . 3 |- (((B e. fBas /\ F e. Fil /\ X = Y) /\ B C_ F) -> (filGen` B) C_ F)
23	22	ex 402	. 2 |- ((B e. fBas /\ F e. Fil /\ X = Y) -> (B C_ F -> (filGen` B) C_ F))
24		fbssfg 10285	. . . 4 |- (B e. fBas -> B C_ (filGen` B))
25		sstr2 2623	. . . 4 |- (B C_ (filGen` B) -> ((filGen` B) C_ F -> B C_ F))
26	24, 25	syl 12	. . 3 |- (B e. fBas -> ((filGen` B) C_ F -> B C_ F))
27	26	3ad2ant1 897	. 2 |- ((B e. fBas /\ F e. Fil /\ X = Y) -> ((filGen` B) C_ F -> B C_ F))
28	23, 27	impbid 574	1 |- ((B e. fBas /\ F e. Fil /\ X = Y) -> (B C_ F <-> (filGen` B) C_ F))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106   C_ wss 2593  U.cuni 3177  ` cfv 3998  fBascfbas 10257  filGencfg 10258  Filcfil 10264

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-fg 10260  df-fil 10265

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