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Theorem fgtr 21604
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) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) = 𝐹)

Proof of Theorem fgtr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filfbas 21562 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbncp 21553 . . . . . . . 8 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ (𝑋𝐴) ∈ 𝐹)
31, 2sylan 488 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ¬ (𝑋𝐴) ∈ 𝐹)
4 filelss 21566 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
5 trfil3 21602 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋𝐴) ∈ 𝐹))
64, 5syldan 487 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ((𝐹t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋𝐴) ∈ 𝐹))
73, 6mpbird 247 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (Fil‘𝐴))
8 filfbas 21562 . . . . . 6 ((𝐹t 𝐴) ∈ (Fil‘𝐴) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
97, 8syl 17 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
10 restsspw 16013 . . . . . 6 (𝐹t 𝐴) ⊆ 𝒫 𝐴
11 sspwb 4878 . . . . . . 7 (𝐴𝑋 ↔ 𝒫 𝐴 ⊆ 𝒫 𝑋)
124, 11sylib 208 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
1310, 12syl5ss 3594 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝒫 𝑋)
14 filtop 21569 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
1514adantr 481 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝑋𝐹)
16 fbasweak 21579 . . . . 5 (((𝐹t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹t 𝐴) ⊆ 𝒫 𝑋𝑋𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝑋))
179, 13, 15, 16syl3anc 1323 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝑋))
181adantr 481 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐹 ∈ (fBas‘𝑋))
19 trfilss 21603 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)
20 fgss 21587 . . . 4 (((𝐹t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ (𝑋filGen𝐹))
2117, 18, 19, 20syl3anc 1323 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ (𝑋filGen𝐹))
22 fgfil 21589 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
2322adantr 481 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen𝐹) = 𝐹)
2421, 23sseqtrd 3620 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ 𝐹)
25 filelss 21566 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
2625ex 450 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹𝑥𝑋))
2726adantr 481 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹𝑥𝑋))
28 elrestr 16010 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝑥𝐹) → (𝑥𝐴) ∈ (𝐹t 𝐴))
29283expa 1262 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → (𝑥𝐴) ∈ (𝐹t 𝐴))
30 inss1 3811 . . . . . . 7 (𝑥𝐴) ⊆ 𝑥
31 sseq1 3605 . . . . . . . 8 (𝑦 = (𝑥𝐴) → (𝑦𝑥 ↔ (𝑥𝐴) ⊆ 𝑥))
3231rspcev 3295 . . . . . . 7 (((𝑥𝐴) ∈ (𝐹t 𝐴) ∧ (𝑥𝐴) ⊆ 𝑥) → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)
3329, 30, 32sylancl 693 . . . . . 6 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)
3433ex 450 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹 → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥))
3527, 34jcad 555 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹 → (𝑥𝑋 ∧ ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)))
36 elfg 21585 . . . . 5 ((𝐹t 𝐴) ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen(𝐹t 𝐴)) ↔ (𝑥𝑋 ∧ ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)))
3717, 36syl 17 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥 ∈ (𝑋filGen(𝐹t 𝐴)) ↔ (𝑥𝑋 ∧ ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)))
3835, 37sylibrd 249 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹𝑥 ∈ (𝑋filGen(𝐹t 𝐴))))
3938ssrdv 3589 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹t 𝐴)))
4024, 39eqssd 3600 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  cdif 3552  cin 3554  wss 3555  𝒫 cpw 4130  cfv 5847  (class class class)co 6604  t crest 16002  fBascfbas 19653  filGencfg 19654  Filcfil 21559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-rest 16004  df-fbas 19662  df-fg 19663  df-fil 21560
This theorem is referenced by:  cfilres  23002
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