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Theorem fgtr 23898
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 23856 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbncp 23847 . . . . . . . 8 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ (𝑋𝐴) ∈ 𝐹)
31, 2sylan 580 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ¬ (𝑋𝐴) ∈ 𝐹)
4 filelss 23860 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
5 trfil3 23896 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋𝐴) ∈ 𝐹))
64, 5syldan 591 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ((𝐹t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋𝐴) ∈ 𝐹))
73, 6mpbird 257 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (Fil‘𝐴))
8 filfbas 23856 . . . . . 6 ((𝐹t 𝐴) ∈ (Fil‘𝐴) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
97, 8syl 17 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
10 restsspw 17476 . . . . . 6 (𝐹t 𝐴) ⊆ 𝒫 𝐴
114sspwd 4613 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
1210, 11sstrid 3995 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝒫 𝑋)
13 filtop 23863 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
1413adantr 480 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝑋𝐹)
15 fbasweak 23873 . . . . 5 (((𝐹t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹t 𝐴) ⊆ 𝒫 𝑋𝑋𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝑋))
169, 12, 14, 15syl3anc 1373 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝑋))
171adantr 480 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐹 ∈ (fBas‘𝑋))
18 trfilss 23897 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)
19 fgss 23881 . . . 4 (((𝐹t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ (𝑋filGen𝐹))
2016, 17, 18, 19syl3anc 1373 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ (𝑋filGen𝐹))
21 fgfil 23883 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
2221adantr 480 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen𝐹) = 𝐹)
2320, 22sseqtrd 4020 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ 𝐹)
24 filelss 23860 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
2524ex 412 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹𝑥𝑋))
2625adantr 480 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹𝑥𝑋))
27 elrestr 17473 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝑥𝐹) → (𝑥𝐴) ∈ (𝐹t 𝐴))
28273expa 1119 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → (𝑥𝐴) ∈ (𝐹t 𝐴))
29 inss1 4237 . . . . . . 7 (𝑥𝐴) ⊆ 𝑥
30 sseq1 4009 . . . . . . . 8 (𝑦 = (𝑥𝐴) → (𝑦𝑥 ↔ (𝑥𝐴) ⊆ 𝑥))
3130rspcev 3622 . . . . . . 7 (((𝑥𝐴) ∈ (𝐹t 𝐴) ∧ (𝑥𝐴) ⊆ 𝑥) → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)
3228, 29, 31sylancl 586 . . . . . 6 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)
3332ex 412 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹 → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥))
3426, 33jcad 512 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹 → (𝑥𝑋 ∧ ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)))
35 elfg 23879 . . . . 5 ((𝐹t 𝐴) ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen(𝐹t 𝐴)) ↔ (𝑥𝑋 ∧ ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)))
3616, 35syl 17 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥 ∈ (𝑋filGen(𝐹t 𝐴)) ↔ (𝑥𝑋 ∧ ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)))
3734, 36sylibrd 259 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹𝑥 ∈ (𝑋filGen(𝐹t 𝐴))))
3837ssrdv 3989 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹t 𝐴)))
3923, 38eqssd 4001 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  cin 3950  wss 3951  𝒫 cpw 4600  cfv 6561  (class class class)co 7431  t crest 17465  fBascfbas 21352  filGencfg 21353  Filcfil 23853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-rest 17467  df-fbas 21361  df-fg 21362  df-fil 23854
This theorem is referenced by:  cfilres  25330
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