Theorem fgcl 22488

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 22481	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑧 ∈ (𝑋filGen𝐹) ↔ (𝑧 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)))
2		elfvex 6705	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
3		fbasne0 22440	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
4		n0 4312	. . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐹)
5	3, 4	sylib 220	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦 𝑦 ∈ 𝐹)
6		fbelss 22443	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝑋)
7	6	ex 415	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑦 ∈ 𝐹 → 𝑦 ⊆ 𝑋))
8	7	ancld 553	. . . . . 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 3146	. . . 4 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋 ↔ ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋))
12	10, 11	sylibr 236	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋)
13		elfvdm 6704	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
14		sseq2 3995	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑋))
15	14	rexbidv 3299	. . . . 5 ⊢ (𝑧 = 𝑋 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
16	15	sbcieg 3812	. . . 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 259	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → [𝑋 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)
19		0nelfb 22441	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
20		0ex 5213	. . . . 5 ⊢ ∅ ∈ V
21		sseq2 3995	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ ∅))
22	21	rexbidv 3299	. . . . 5 ⊢ (𝑧 = ∅ → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅))
23	20, 22	sbcie 3814	. . . 4 ⊢ ([∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅)
24		ss0 4354	. . . . . . 7 ⊢ (𝑦 ⊆ ∅ → 𝑦 = ∅)
25	24	eleq1d 2899	. . . . . 6 ⊢ (𝑦 ⊆ ∅ → (𝑦 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
26	25	biimpac 481	. . . . 5 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ ∅) → ∅ ∈ 𝐹)
27	26	rexlimiva 3283	. . . 4 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅ → ∅ ∈ 𝐹)
28	23, 27	sylbi 219	. . 3 ⊢ ([∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 → ∅ ∈ 𝐹)
29	19, 28	nsyl 142	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ [∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)
30		sstr 3977	. . . . . 6 ⊢ ((𝑦 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑢) → 𝑦 ⊆ 𝑢)
31	30	expcom 416	. . . . 5 ⊢ (𝑣 ⊆ 𝑢 → (𝑦 ⊆ 𝑣 → 𝑦 ⊆ 𝑢))
32	31	reximdv 3275	. . . 4 ⊢ (𝑣 ⊆ 𝑢 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
33	32	3ad2ant3 1131	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑢) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
34		vex 3499	. . . 4 ⊢ 𝑣 ∈ V
35		sseq2 3995	. . . . 5 ⊢ (𝑧 = 𝑣 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑣))
36	35	rexbidv 3299	. . . 4 ⊢ (𝑧 = 𝑣 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣))
37	34, 36	sbcie 3814	. . 3 ⊢ ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣)
38		vex 3499	. . . 4 ⊢ 𝑢 ∈ V
39		sseq2 3995	. . . . 5 ⊢ (𝑧 = 𝑢 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑢))
40	39	rexbidv 3299	. . . 4 ⊢ (𝑧 = 𝑢 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
41	38, 40	sbcie 3814	. . 3 ⊢ ([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢)
42	33, 37, 41	3imtr4g 298	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑢) → ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 → [𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧))
43		fbasssin 22446	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹 ∧ 𝑤 ∈ 𝐹) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤))
44	43	3expib 1118	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝑧 ∈ 𝐹 ∧ 𝑤 ∈ 𝐹) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤)))
45		sstr2 3976	. . . . . . . . . . . . . 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 3275	. . . . . . . . . . . 12 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
48		ss2in 4215	. . . . . . . . . . . 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 447	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑧 ∈ 𝐹 → (𝑤 ∈ 𝐹 → (𝑧 ⊆ 𝑢 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))))
52	51	imp31 420	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑤 ∈ 𝐹) → (𝑧 ⊆ 𝑢 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
53	52	impancom 454	. . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑧 ⊆ 𝑢) → (𝑤 ∈ 𝐹 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
54	53	rexlimdv 3285	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑧 ⊆ 𝑢) → (∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
55	54	rexlimdva2 3289	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 → (∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
56	55	impd 413	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
57	56	3ad2ant1 1129	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑋) → ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
58		sseq1 3994	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑢))
59	58	cbvrexvw 3452	. . . . 5 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢)
60	41, 59	bitri 277	. . . 4 ⊢ ([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢)
61		sseq1 3994	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ⊆ 𝑣 ↔ 𝑤 ⊆ 𝑣))
62	61	cbvrexvw 3452	. . . . 5 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 ↔ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣)
63	37, 62	bitri 277	. . . 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 5223	. . . 4 ⊢ (𝑢 ∩ 𝑣) ∈ V
66		sseq2 3995	. . . . 5 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑢 ∩ 𝑣)))
67	66	rexbidv 3299	. . . 4 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
68	65, 67	sbcie 3814	. . 3 ⊢ ([(𝑢 ∩ 𝑣) / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))
69	57, 64, 68	3imtr4g 298	. 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 22468	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141  [wsbc 3774   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  dom cdm 5557  ‘cfv 6357  (class class class)co 7158  fBascfbas 20535  filGencfg 20536  Filcfil 22455

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-fbas 20544  df-fg 20545  df-fil 22456

This theorem is referenced by:  fgabs  22489  trfg  22501  isufil2  22518  ssufl  22528  ufileu  22529  filufint  22530  fixufil  22532  uffixfr  22533  fmfil  22554  fmfg  22559  elfm3  22560  rnelfm  22563  fmfnfmlem2  22565  fmfnfm  22568  fbflim  22586  hausflim  22591  flimclslem  22594  flffbas  22605  fclsbas  22631  fclsfnflim  22637  flimfnfcls  22638  fclscmp  22640  haustsms  22746  tsmscls  22748  tsmsmhm  22756  tsmsadd  22757  cfilufg  22904  metust  23170  fgcfil  23876  cmetcaulem  23893  cmetss  23921  minveclem4a  24035  minveclem4  24037