Theorem fgtr 23278

Description: If 𝐴 is a member of the filter, then truncating 𝐹 to 𝐴 and regenerating the behavior outside 𝐴 using filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
fgtr	⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) = 𝐹)

Proof of Theorem fgtr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		filfbas 23236	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbncp 23227	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
3	1, 2	sylan 580	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
4		filelss 23240	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
5		trfil3 23276	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
6	4, 5	syldan 591	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
7	3, 6	mpbird 256	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (Fil‘𝐴))
8		filfbas 23236	. . . . . 6 ⊢ ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
9	7, 8	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
10		restsspw 17327	. . . . . 6 ⊢ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝐴
11	4	sspwd 4578	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
12	10, 11	sstrid 3958	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋)
13		filtop 23243	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
14	13	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝑋 ∈ 𝐹)
15		fbasweak 23253	. . . . 5 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
16	9, 12, 14, 15	syl3anc 1371	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
17	1	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
18		trfilss 23277	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝐹)
19		fgss 23261	. . . 4 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹 ↾t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
20	16, 17, 18, 19	syl3anc 1371	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
21		fgfil 23263	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
22	21	adantr 481	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen𝐹) = 𝐹)
23	20, 22	sseqtrd 3987	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ 𝐹)
24		filelss 23240	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
25	24	ex 413	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
26	25	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
27		elrestr 17324	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
28	27	3expa 1118	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
29		inss1 4193	. . . . . . 7 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
30		sseq1 3972	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐴) → (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝐴) ⊆ 𝑥))
31	30	rspcev 3582	. . . . . . 7 ⊢ (((𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ (𝑥 ∩ 𝐴) ⊆ 𝑥) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
32	28, 29, 31	sylancl 586	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
33	32	ex 413	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥))
34	26, 33	jcad 513	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
35		elfg 23259	. . . . 5 ⊢ ((𝐹 ↾t 𝐴) ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴)) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
36	16, 35	syl 17	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴)) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
37	34, 36	sylibrd 258	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴))))
38	37	ssrdv 3953	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹 ↾t 𝐴)))
39	23, 38	eqssd 3964	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3069   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4565  ‘cfv 6501  (class class class)co 7362   ↾_t crest 17316  fBascfbas 20821  filGencfg 20822  Filcfil 23233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-rest 17318  df-fbas 20830  df-fg 20831  df-fil 23234

This theorem is referenced by:  cfilres  24697