Theorem fgcl 23820

Description: A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fgcl	⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))

Proof of Theorem fgcl

Dummy variables 𝑣 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfg 23813	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑧 ∈ (𝑋filGen𝐹) ↔ (𝑧 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)))
2		elfvex 6867	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
3		fbasne0 23772	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
4		n0 4303	. . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐹)
5	3, 4	sylib 218	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦 𝑦 ∈ 𝐹)
6		fbelss 23775	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝑋)
7	6	ex 412	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑦 ∈ 𝐹 → 𝑦 ⊆ 𝑋))
8	7	ancld 550	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑦 ∈ 𝐹 → (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋)))
9	8	eximdv 1918	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∃𝑦 𝑦 ∈ 𝐹 → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋)))
10	5, 9	mpd 15	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋))
11		df-rex 3059	. . . 4 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋 ↔ ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋))
12	10, 11	sylibr 234	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋)
13		elfvdm 6866	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
14		sseq2 3958	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑋))
15	14	rexbidv 3158	. . . . 5 ⊢ (𝑧 = 𝑋 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
16	15	sbcieg 3778	. . . 4 ⊢ (𝑋 ∈ dom fBas → ([𝑋 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
17	13, 16	syl 17	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ([𝑋 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
18	12, 17	mpbird 257	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → [𝑋 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)
19		0nelfb 23773	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
20		0ex 5250	. . . . 5 ⊢ ∅ ∈ V
21		sseq2 3958	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ ∅))
22	21	rexbidv 3158	. . . . 5 ⊢ (𝑧 = ∅ → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅))
23	20, 22	sbcie 3780	. . . 4 ⊢ ([∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅)
24		ss0 4352	. . . . . . 7 ⊢ (𝑦 ⊆ ∅ → 𝑦 = ∅)
25	24	eleq1d 2819	. . . . . 6 ⊢ (𝑦 ⊆ ∅ → (𝑦 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
26	25	biimpac 478	. . . . 5 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ ∅) → ∅ ∈ 𝐹)
27	26	rexlimiva 3127	. . . 4 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅ → ∅ ∈ 𝐹)
28	23, 27	sylbi 217	. . 3 ⊢ ([∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 → ∅ ∈ 𝐹)
29	19, 28	nsyl 140	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ [∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)
30		sstr 3940	. . . . . 6 ⊢ ((𝑦 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑢) → 𝑦 ⊆ 𝑢)
31	30	expcom 413	. . . . 5 ⊢ (𝑣 ⊆ 𝑢 → (𝑦 ⊆ 𝑣 → 𝑦 ⊆ 𝑢))
32	31	reximdv 3149	. . . 4 ⊢ (𝑣 ⊆ 𝑢 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
33	32	3ad2ant3 1135	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑢) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
34		vex 3442	. . . 4 ⊢ 𝑣 ∈ V
35		sseq2 3958	. . . . 5 ⊢ (𝑧 = 𝑣 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑣))
36	35	rexbidv 3158	. . . 4 ⊢ (𝑧 = 𝑣 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣))
37	34, 36	sbcie 3780	. . 3 ⊢ ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣)
38		vex 3442	. . . 4 ⊢ 𝑢 ∈ V
39		sseq2 3958	. . . . 5 ⊢ (𝑧 = 𝑢 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑢))
40	39	rexbidv 3158	. . . 4 ⊢ (𝑧 = 𝑢 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
41	38, 40	sbcie 3780	. . 3 ⊢ ([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢)
42	33, 37, 41	3imtr4g 296	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑢) → ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 → [𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧))
43		fbasssin 23778	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹 ∧ 𝑤 ∈ 𝐹) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤))
44	43	3expib 1122	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝑧 ∈ 𝐹 ∧ 𝑤 ∈ 𝐹) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤)))
45		sstr2 3938	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ (𝑧 ∩ 𝑤) → ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → 𝑦 ⊆ (𝑢 ∩ 𝑣)))
46	45	com12 32	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (𝑦 ⊆ (𝑧 ∩ 𝑤) → 𝑦 ⊆ (𝑢 ∩ 𝑣)))
47	46	reximdv 3149	. . . . . . . . . . . 12 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
48		ss2in 4195	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣))
49	47, 48	syl11 33	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤) → ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
50	44, 49	syl6 35	. . . . . . . . . 10 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝑧 ∈ 𝐹 ∧ 𝑤 ∈ 𝐹) → ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
51	50	exp5c 444	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑧 ∈ 𝐹 → (𝑤 ∈ 𝐹 → (𝑧 ⊆ 𝑢 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))))
52	51	imp31 417	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑤 ∈ 𝐹) → (𝑧 ⊆ 𝑢 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
53	52	impancom 451	. . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑧 ⊆ 𝑢) → (𝑤 ∈ 𝐹 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
54	53	rexlimdv 3133	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑧 ⊆ 𝑢) → (∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
55	54	rexlimdva2 3137	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 → (∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
56	55	impd 410	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
57	56	3ad2ant1 1133	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑋) → ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
58		sseq1 3957	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑢))
59	58	cbvrexvw 3213	. . . . 5 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢)
60	41, 59	bitri 275	. . . 4 ⊢ ([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢)
61		sseq1 3957	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ⊆ 𝑣 ↔ 𝑤 ⊆ 𝑣))
62	61	cbvrexvw 3213	. . . . 5 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 ↔ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣)
63	37, 62	bitri 275	. . . 4 ⊢ ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣)
64	60, 63	anbi12i 628	. . 3 ⊢ (([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ∧ [𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧) ↔ (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣))
65	38	inex1 5260	. . . 4 ⊢ (𝑢 ∩ 𝑣) ∈ V
66		sseq2 3958	. . . . 5 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑢 ∩ 𝑣)))
67	66	rexbidv 3158	. . . 4 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
68	65, 67	sbcie 3780	. . 3 ⊢ ([(𝑢 ∩ 𝑣) / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))
69	57, 64, 68	3imtr4g 296	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑋) → (([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ∧ [𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧) → [(𝑢 ∩ 𝑣) / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧))
70	1, 2, 18, 29, 42, 69	isfild 23800	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058  Vcvv 3438  [wsbc 3738   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  dom cdm 5622  ‘cfv 6490  (class class class)co 7356  fBascfbas 21295  filGencfg 21296  Filcfil 23787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-fbas 21304  df-fg 21305  df-fil 23788

This theorem is referenced by:  fgabs  23821  trfg  23833  isufil2  23850  ssufl  23860  ufileu  23861  filufint  23862  fixufil  23864  uffixfr  23865  fmfil  23886  fmfg  23891  elfm3  23892  rnelfm  23895  fmfnfmlem2  23897  fmfnfm  23900  fbflim  23918  hausflim  23923  flimclslem  23926  flffbas  23937  fclsbas  23963  fclsfnflim  23969  flimfnfcls  23970  fclscmp  23972  haustsms  24078  tsmscls  24080  tsmsmhm  24088  tsmsadd  24089  cfilufg  24234  metust  24500  fgcfil  25225  cmetcaulem  25242  cmetss  25270  minveclem4a  25384  minveclem4  25386