Theorem fgtr 22498

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 22456	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbncp 22447	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
3	1, 2	sylan 582	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝑋 ∖ 𝐴) ∈ 𝐹)
4		filelss 22460	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
5		trfil3 22496	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
6	4, 5	syldan 593	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝐹 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋 ∖ 𝐴) ∈ 𝐹))
7	3, 6	mpbird 259	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (Fil‘𝐴))
8		filfbas 22456	. . . . . 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 16705	. . . . . 6 ⊢ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝐴
11	4	sspwd 4554	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
12	10, 11	sstrid 3978	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋)
13		filtop 22463	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
14	13	adantr 483	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝑋 ∈ 𝐹)
15		fbasweak 22473	. . . . 5 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹 ↾t 𝐴) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
16	9, 12, 14, 15	syl3anc 1367	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝑋))
17	1	adantr 483	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
18		trfilss 22497	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝐹)
19		fgss 22481	. . . 4 ⊢ (((𝐹 ↾t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹 ↾t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
20	16, 17, 18, 19	syl3anc 1367	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ (𝑋filGen𝐹))
21		fgfil 22483	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
22	21	adantr 483	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen𝐹) = 𝐹)
23	20, 22	sseqtrd 4007	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) ⊆ 𝐹)
24		filelss 22460	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
25	24	ex 415	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
26	25	adantr 483	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ 𝑋))
27		elrestr 16702	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
28	27	3expa 1114	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴))
29		inss1 4205	. . . . . . 7 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
30		sseq1 3992	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐴) → (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝐴) ⊆ 𝑥))
31	30	rspcev 3623	. . . . . . 7 ⊢ (((𝑥 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ (𝑥 ∩ 𝐴) ⊆ 𝑥) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
32	28, 29, 31	sylancl 588	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)
33	32	ex 415	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥))
34	26, 33	jcad 515	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝐹 ↾t 𝐴)𝑦 ⊆ 𝑥)))
35		elfg 22479	. . . . 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 261	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝑋filGen(𝐹 ↾t 𝐴))))
38	37	ssrdv 3973	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹 ↾t 𝐴)))
39	23, 38	eqssd 3984	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  ‘cfv 6355  (class class class)co 7156   ↾_t crest 16694  fBascfbas 20533  filGencfg 20534  Filcfil 22453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-rest 16696  df-fbas 20542  df-fg 20543  df-fil 22454

This theorem is referenced by:  cfilres  23899