Theorem fgtr 23833

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 23791	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbncp 23782	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
3	1, 2	sylan 581	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
4		filelss 23795	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
5		trfil3 23831	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
6	4, 5	syldan 592	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
7	3, 6	mpbird 257	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (Fil‘𝐴))
8		filfbas 23791	. . . . . 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 17352	. . . . . 6 ⊢ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝐴
11	4	sspwd 4555	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
12	10, 11	sstrid 3934	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋)
13		filtop 23798	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
14	13	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝑋 ∈ 𝐹)
15		fbasweak 23808	. . . . 5 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
16	9, 12, 14, 15	syl3anc 1374	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
17	1	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
18		trfilss 23832	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝐹)
19		fgss 23816	. . . 4 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹 ↾t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
20	16, 17, 18, 19	syl3anc 1374	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
21		fgfil 23818	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
22	21	adantr 480	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen𝐹) = 𝐹)
23	20, 22	sseqtrd 3959	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ 𝐹)
24		filelss 23795	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
25	24	ex 412	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
26	25	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
27		elrestr 17349	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
28	27	3expa 1119	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
29		inss1 4178	. . . . . . 7 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
30		sseq1 3948	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐴) → (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝐴) ⊆ 𝑥))
31	30	rspcev 3565	. . . . . . 7 ⊢ (((𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ (𝑥 ∩ 𝐴) ⊆ 𝑥) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
32	28, 29, 31	sylancl 587	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
33	32	ex 412	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥))
34	26, 33	jcad 512	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
35		elfg 23814	. . . . 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 259	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴))))
38	37	ssrdv 3928	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹 ↾t 𝐴)))
39	23, 38	eqssd 3940	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ‘cfv 6490  (class class class)co 7358   ↾_t crest 17341  fBascfbas 21299  filGencfg 21300  Filcfil 23788

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-rest 17343  df-fbas 21308  df-fg 21309  df-fil 23789

This theorem is referenced by:  cfilres  25241