Theorem fgcl 23907

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 23900	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑧 ∈ (𝑋filGen𝐹) ↔ (𝑧 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)))
2		elfvex 6958	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
3		fbasne0 23859	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
4		n0 4376	. . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐹)
5	3, 4	sylib 218	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦 𝑦 ∈ 𝐹)
6		fbelss 23862	. . . . . . . 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 1916	. . . . 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 3077	. . . 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 6957	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
14		sseq2 4035	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑋))
15	14	rexbidv 3185	. . . . 5 ⊢ (𝑧 = 𝑋 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
16	15	sbcieg 3845	. . . 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 23860	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
20		0ex 5325	. . . . 5 ⊢ ∅ ∈ V
21		sseq2 4035	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ ∅))
22	21	rexbidv 3185	. . . . 5 ⊢ (𝑧 = ∅ → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅))
23	20, 22	sbcie 3848	. . . 4 ⊢ ([∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅)
24		ss0 4425	. . . . . . 7 ⊢ (𝑦 ⊆ ∅ → 𝑦 = ∅)
25	24	eleq1d 2829	. . . . . 6 ⊢ (𝑦 ⊆ ∅ → (𝑦 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
26	25	biimpac 478	. . . . 5 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ ∅) → ∅ ∈ 𝐹)
27	26	rexlimiva 3153	. . . 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 4017	. . . . . 6 ⊢ ((𝑦 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑢) → 𝑦 ⊆ 𝑢)
31	30	expcom 413	. . . . 5 ⊢ (𝑣 ⊆ 𝑢 → (𝑦 ⊆ 𝑣 → 𝑦 ⊆ 𝑢))
32	31	reximdv 3176	. . . 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 3492	. . . 4 ⊢ 𝑣 ∈ V
35		sseq2 4035	. . . . 5 ⊢ (𝑧 = 𝑣 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑣))
36	35	rexbidv 3185	. . . 4 ⊢ (𝑧 = 𝑣 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣))
37	34, 36	sbcie 3848	. . 3 ⊢ ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣)
38		vex 3492	. . . 4 ⊢ 𝑢 ∈ V
39		sseq2 4035	. . . . 5 ⊢ (𝑧 = 𝑢 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑢))
40	39	rexbidv 3185	. . . 4 ⊢ (𝑧 = 𝑢 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
41	38, 40	sbcie 3848	. . 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 23865	. . . . . . . . . . . 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 4015	. . . . . . . . . . . . . 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 3176	. . . . . . . . . . . 12 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
48		ss2in 4266	. . . . . . . . . . . 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 3159	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑧 ⊆ 𝑢) → (∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
55	54	rexlimdva2 3163	. . . . 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 4034	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑢))
59	58	cbvrexvw 3244	. . . . 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 4034	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ⊆ 𝑣 ↔ 𝑤 ⊆ 𝑣))
62	61	cbvrexvw 3244	. . . . 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 627	. . 3 ⊢ (([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ∧ [𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧) ↔ (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣))
65	38	inex1 5335	. . . 4 ⊢ (𝑢 ∩ 𝑣) ∈ V
66		sseq2 4035	. . . . 5 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑢 ∩ 𝑣)))
67	66	rexbidv 3185	. . . 4 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
68	65, 67	sbcie 3848	. . 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 23887	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  Vcvv 3488  [wsbc 3804   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  fBascfbas 21375  filGencfg 21376  Filcfil 23874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-fbas 21384  df-fg 21385  df-fil 23875

This theorem is referenced by:  fgabs  23908  trfg  23920  isufil2  23937  ssufl  23947  ufileu  23948  filufint  23949  fixufil  23951  uffixfr  23952  fmfil  23973  fmfg  23978  elfm3  23979  rnelfm  23982  fmfnfmlem2  23984  fmfnfm  23987  fbflim  24005  hausflim  24010  flimclslem  24013  flffbas  24024  fclsbas  24050  fclsfnflim  24056  flimfnfcls  24057  fclscmp  24059  haustsms  24165  tsmscls  24167  tsmsmhm  24175  tsmsadd  24176  cfilufg  24323  metust  24592  fgcfil  25324  cmetcaulem  25341  cmetss  25369  minveclem4a  25483  minveclem4  25485