Theorem fgcl 24209

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 24202	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑧 ∈ (𝑋filGen𝐹) ↔ (𝑧 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)))
2		elfvex 6928	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ V)
3		fbasne0 24161	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
4		n0 4302	. . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐹)
5	3, 4	sylib 221	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦 𝑦 ∈ 𝐹)
6		fbelss 24164	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝑋)
7	6	ex 418	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑦 ∈ 𝐹 → 𝑦 ⊆ 𝑋))
8	7	ancld 560	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑦 ∈ 𝐹 → (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋)))
9	8	eximdv 1951	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∃𝑦 𝑦 ∈ 𝐹 → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋)))
10	5, 9	mpd 16	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋))
11		df-rex 3090	. . . 4 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋 ↔ ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋))
12	10, 11	sylibr 237	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋)
13		elfvdm 6927	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
14		sseq2 3938	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑋))
15	14	rexbidv 3189	. . . . 5 ⊢ (𝑧 = 𝑋 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
16	15	sbcieg 3780	. . . 4 ⊢ (𝑋 ∈ dom fBas → ([𝑋 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
17	13, 16	syl 18	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ([𝑋 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑋))
18	12, 17	mpbird 260	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → [𝑋 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)
19		0nelfb 24162	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
20		0ex 5263	. . . . 5 ⊢ ∅ ∈ V
21		sseq2 3938	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ ∅))
22	21	rexbidv 3189	. . . . 5 ⊢ (𝑧 = ∅ → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅))
23	20, 22	sbcie 3782	. . . 4 ⊢ ([∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅)
24		ss0 4354	. . . . . . 7 ⊢ (𝑦 ⊆ ∅ → 𝑦 = ∅)
25	24	eleq1d 2848	. . . . . 6 ⊢ (𝑦 ⊆ ∅ → (𝑦 ∈ 𝐹 ↔ ∅ ∈ 𝐹))
26	25	biimpac 484	. . . . 5 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ ∅) → ∅ ∈ 𝐹)
27	26	rexlimiva 3158	. . . 4 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ ∅ → ∅ ∈ 𝐹)
28	23, 27	sylbi 220	. . 3 ⊢ ([∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 → ∅ ∈ 𝐹)
29	19, 28	nsyl 141	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ [∅ / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧)
30		sstr 3920	. . . . . 6 ⊢ ((𝑦 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑢) → 𝑦 ⊆ 𝑢)
31	30	expcom 419	. . . . 5 ⊢ (𝑣 ⊆ 𝑢 → (𝑦 ⊆ 𝑣 → 𝑦 ⊆ 𝑢))
32	31	reximdv 3180	. . . 4 ⊢ (𝑣 ⊆ 𝑢 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
33	32	3ad2ant3 1153	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑢) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
34		vex 3457	. . . 4 ⊢ 𝑣 ∈ V
35		sseq2 3938	. . . . 5 ⊢ (𝑧 = 𝑣 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑣))
36	35	rexbidv 3189	. . . 4 ⊢ (𝑧 = 𝑣 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣))
37	34, 36	sbcie 3782	. . 3 ⊢ ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣)
38		vex 3457	. . . 4 ⊢ 𝑢 ∈ V
39		sseq2 3938	. . . . 5 ⊢ (𝑧 = 𝑢 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑢))
40	39	rexbidv 3189	. . . 4 ⊢ (𝑧 = 𝑢 → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
41	38, 40	sbcie 3782	. . 3 ⊢ ([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢)
42	33, 37, 41	3imtr4g 299	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑢) → ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 → [𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧))
43		fbasssin 24167	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹 ∧ 𝑤 ∈ 𝐹) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤))
44	43	3expib 1140	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝑧 ∈ 𝐹 ∧ 𝑤 ∈ 𝐹) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤)))
45		sstr2 3919	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ (𝑧 ∩ 𝑤) → ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → 𝑦 ⊆ (𝑢 ∩ 𝑣)))
46	45	com12 33	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (𝑦 ⊆ (𝑧 ∩ 𝑤) → 𝑦 ⊆ (𝑢 ∩ 𝑣)))
47	46	reximdv 3180	. . . . . . . . . . . 12 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
48		ss2in 4143	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣))
49	47, 48	syl11 34	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑧 ∩ 𝑤) → ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
50	44, 49	syl6 36	. . . . . . . . . 10 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝑧 ∈ 𝐹 ∧ 𝑤 ∈ 𝐹) → ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
51	50	exp5c 450	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑧 ∈ 𝐹 → (𝑤 ∈ 𝐹 → (𝑧 ⊆ 𝑢 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))))
52	51	imp31 423	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑤 ∈ 𝐹) → (𝑧 ⊆ 𝑢 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
53	52	impancom 457	. . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑧 ⊆ 𝑢) → (𝑤 ∈ 𝐹 → (𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
54	53	rexlimdv 3164	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑧 ∈ 𝐹) ∧ 𝑧 ⊆ 𝑢) → (∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
55	54	rexlimdva2 3168	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 → (∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣 → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))))
56	55	impd 416	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
57	56	3ad2ant1 1151	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ⊆ 𝑋 ∧ 𝑣 ⊆ 𝑋) → ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
58		sseq1 3937	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑢))
59	58	cbvrexvw 3244	. . . . 5 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢)
60	41, 59	bitri 278	. . . 4 ⊢ ([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢)
61		sseq1 3937	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ⊆ 𝑣 ↔ 𝑤 ⊆ 𝑣))
62	61	cbvrexvw 3244	. . . . 5 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑣 ↔ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣)
63	37, 62	bitri 278	. . . 4 ⊢ ([𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣)
64	60, 63	anbi12i 640	. . 3 ⊢ (([𝑢 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ∧ [𝑣 / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧) ↔ (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑢 ∧ ∃𝑤 ∈ 𝐹 𝑤 ⊆ 𝑣))
65	38	inex1 5280	. . . 4 ⊢ (𝑢 ∩ 𝑣) ∈ V
66		sseq2 3938	. . . . 5 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑢 ∩ 𝑣)))
67	66	rexbidv 3189	. . . 4 ⊢ (𝑧 = (𝑢 ∩ 𝑣) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣)))
68	65, 67	sbcie 3782	. . 3 ⊢ ([(𝑢 ∩ 𝑣) / 𝑧]∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑧 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ (𝑢 ∩ 𝑣))
69	57, 64, 68	3imtr4g 299	. 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 24189	1 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2958  ∃wrex 3089  Vcvv 3453  [wsbc 3741   ⊆ wss 3899   ∩ cin 4091  ∅c0 4280  dom cdm 5654  ‘cfv 6546  (class class class)co 7427  fBascfbas 21678  filGencfg 21679  Filcfil 24176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-fbas 21687  df-fg 21688  df-fil 24177

This theorem is used by:  fgabs  24210  trfg  24222  isufil2  24239  ssufl  24249  ufileu  24250  filufint  24251  fixufil  24253  uffixfr  24254  fmfil  24275  fmfg  24280  elfm3  24281  rnelfm  24284  fmfnfmlem2  24286  fmfnfm  24289  fbflim  24307  hausflim  24312  flimclslem  24315  flffbas  24326  fclsbas  24352  fclsfnflim  24358  flimfnfcls  24359  fclscmp  24361  haustsms  24467  tsmscls  24469  tsmsmhm  24477  tsmsadd  24478  cfilufg  24623  metust  24889  fgcfil  25604  cmetcaulem  25621  cmetss  25649  minveclem4a  25763  minveclem4  25765