Theorem fgtr 21504

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 21462	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbncp 21453	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
3	1, 2	sylan 487	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
4		filelss 21466	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
5		trfil3 21502	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
6	4, 5	syldan 486	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
7	3, 6	mpbird 246	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (Fil‘𝐴))
8		filfbas 21462	. . . . . 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 15915	. . . . . 6 ⊢ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝐴
11		sspwb 4844	. . . . . . 7 ⊢ (𝐴 ⊆ 𝑋 ↔ 𝒫 𝐴 ⊆ 𝒫 𝑋)
12	4, 11	sylib 207	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
13	10, 12	syl5ss 3579	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋)
14		filtop 21469	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
15	14	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝑋 ∈ 𝐹)
16		fbasweak 21479	. . . . 5 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
17	9, 13, 15, 16	syl3anc 1318	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
18	1	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
19		trfilss 21503	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝐹)
20		fgss 21487	. . . 4 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹 ↾t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
21	17, 18, 19, 20	syl3anc 1318	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
22		fgfil 21489	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
23	22	adantr 480	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen𝐹) = 𝐹)
24	21, 23	sseqtrd 3604	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ 𝐹)
25		filelss 21466	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
26	25	ex 449	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
27	26	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
28		elrestr 15912	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
29	28	3expa 1257	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
30		inss1 3795	. . . . . . 7 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
31		sseq1 3589	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐴) → (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝐴) ⊆ 𝑥))
32	31	rspcev 3282	. . . . . . 7 ⊢ (((𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ (𝑥 ∩ 𝐴) ⊆ 𝑥) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
33	29, 30, 32	sylancl 693	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
34	33	ex 449	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥))
35	27, 34	jcad 554	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
36		elfg 21485	. . . . 5 ⊢ ((𝐹 ↾t 𝐴) ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴)) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
37	17, 36	syl 17	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴)) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
38	35, 37	sylibrd 248	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴))))
39	38	ssrdv 3574	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹 ↾t 𝐴)))
40	24, 39	eqssd 3585	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  fBascfbas 19555  filGencfg 19556  Filcfil 21459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rest 15906  df-fbas 19564  df-fg 19565  df-fil 21460

This theorem is referenced by:  cfilres  22902