Theorem fgcl 23793

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 23786	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑧 ∈ (𝑋filGen𝐹) ↔ (𝑧 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)))
2		elfvex 6857	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
3		fbasne0 23745	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
4		n0 4300	. . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐹)
5	3, 4	sylib 218	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦 𝑦 ∈ 𝐹)
6		fbelss 23748	. . . . . . . 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 3057	. . . 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 6856	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
14		sseq2 3956	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑋))
15	14	rexbidv 3156	. . . . 5 ⊢ (𝑧 = 𝑋 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
16	15	sbcieg 3776	. . . 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 23746	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
20		0ex 5243	. . . . 5 ⊢ ∅ ∈ V
21		sseq2 3956	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ ∅))
22	21	rexbidv 3156	. . . . 5 ⊢ (𝑧 = ∅ → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅))
23	20, 22	sbcie 3778	. . . 4 ⊢ ([∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅)
24		ss0 4349	. . . . . . 7 ⊢ (𝑦 ⊆ ∅ → 𝑦 = ∅)
25	24	eleq1d 2816	. . . . . 6 ⊢ (𝑦 ⊆ ∅ → (𝑦 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
26	25	biimpac 478	. . . . 5 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ ∅) → ∅ ∈ 𝐹)
27	26	rexlimiva 3125	. . . 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 3938	. . . . . 6 ⊢ ((𝑦 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑢) → 𝑦 ⊆ 𝑢)
31	30	expcom 413	. . . . 5 ⊢ (𝑣 ⊆ 𝑢 → (𝑦 ⊆ 𝑣 → 𝑦 ⊆ 𝑢))
32	31	reximdv 3147	. . . 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 3440	. . . 4 ⊢ 𝑣 ∈ V
35		sseq2 3956	. . . . 5 ⊢ (𝑧 = 𝑣 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑣))
36	35	rexbidv 3156	. . . 4 ⊢ (𝑧 = 𝑣 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣))
37	34, 36	sbcie 3778	. . 3 ⊢ ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣)
38		vex 3440	. . . 4 ⊢ 𝑢 ∈ V
39		sseq2 3956	. . . . 5 ⊢ (𝑧 = 𝑢 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑢))
40	39	rexbidv 3156	. . . 4 ⊢ (𝑧 = 𝑢 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
41	38, 40	sbcie 3778	. . 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 23751	. . . . . . . . . . . 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 3936	. . . . . . . . . . . . . 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 3147	. . . . . . . . . . . 12 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
48		ss2in 4192	. . . . . . . . . . . 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 3131	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑧 ⊆ 𝑢) → (∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
55	54	rexlimdva2 3135	. . . . 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 3955	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑢))
59	58	cbvrexvw 3211	. . . . 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 3955	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ⊆ 𝑣 ↔ 𝑤 ⊆ 𝑣))
62	61	cbvrexvw 3211	. . . . 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 5253	. . . 4 ⊢ (𝑢 ∩ 𝑣) ∈ V
66		sseq2 3956	. . . . 5 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑢 ∩ 𝑣)))
67	66	rexbidv 3156	. . . 4 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
68	65, 67	sbcie 3778	. . 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 23773	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  Vcvv 3436  [wsbc 3736   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280  dom cdm 5614  ‘cfv 6481  (class class class)co 7346  fBascfbas 21279  filGencfg 21280  Filcfil 23760

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-fbas 21288  df-fg 21289  df-fil 23761

This theorem is referenced by:  fgabs  23794  trfg  23806  isufil2  23823  ssufl  23833  ufileu  23834  filufint  23835  fixufil  23837  uffixfr  23838  fmfil  23859  fmfg  23864  elfm3  23865  rnelfm  23868  fmfnfmlem2  23870  fmfnfm  23873  fbflim  23891  hausflim  23896  flimclslem  23899  flffbas  23910  fclsbas  23936  fclsfnflim  23942  flimfnfcls  23943  fclscmp  23945  haustsms  24051  tsmscls  24053  tsmsmhm  24061  tsmsadd  24062  cfilufg  24207  metust  24473  fgcfil  25198  cmetcaulem  25215  cmetss  25243  minveclem4a  25357  minveclem4  25359