Theorem fgtr 24058

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 24016	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbncp 24007	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
3	1, 2	sylan 591	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
4		filelss 24020	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
5		trfil3 24056	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
6	4, 5	syldan 602	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
7	3, 6	mpbird 260	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (Fil‘𝐴))
8		filfbas 24016	. . . . . 6 ⊢ ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
9	7, 8	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴))
10		restsspw 17490	. . . . . 6 ⊢ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝐴
11	4	sspwd 4574	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
12	10, 11	sstrid 3947	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋)
13		filtop 24023	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
14	13	adantr 485	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝑋 ∈ 𝐹)
15		fbasweak 24033	. . . . 5 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
16	9, 12, 14, 15	syl3anc 1397	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
17	1	adantr 485	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
18		trfilss 24057	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝐹)
19		fgss 24041	. . . 4 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹 ↾t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
20	16, 17, 18, 19	syl3anc 1397	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
21		fgfil 24043	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
22	21	adantr 485	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen𝐹) = 𝐹)
23	20, 22	sseqtrd 3972	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ 𝐹)
24		filelss 24020	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
25	24	ex 417	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
26	25	adantr 485	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
27		elrestr 17487	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
28	27	3expa 1135	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
29		inss1 4188	. . . . . . 7 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
30		sseq1 3961	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐴) → (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝐴) ⊆ 𝑥))
31	30	rspcev 3580	. . . . . . 7 ⊢ (((𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ (𝑥 ∩ 𝐴) ⊆ 𝑥) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
32	28, 29, 31	sylancl 597	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
33	32	ex 417	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥))
34	26, 33	jcad 521	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
35		elfg 24039	. . . . 5 ⊢ ((𝐹 ↾t 𝐴) ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴)) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
36	16, 35	syl 18	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴)) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
37	34, 36	sylibrd 262	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴))))
38	37	ssrdv 3942	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹 ↾t 𝐴)))
39	23, 38	eqssd 3953	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) = 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4561  ‘cfv 6536  (class class class)co 7412   ↾_t crest 17479  fBascfbas 21521  filGencfg 21522  Filcfil 24013

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-rest 17481  df-fbas 21530  df-fg 21531  df-fil 24014

This theorem is used by:  cfilres  25466