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Theorem fgtr 23913
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 23871 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbncp 23862 . . . . . . . 8 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ (𝑋𝐴) ∈ 𝐹)
31, 2sylan 580 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ¬ (𝑋𝐴) ∈ 𝐹)
4 filelss 23875 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
5 trfil3 23911 . . . . . . . 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 23871 . . . . . 6 ((𝐹t 𝐴) ∈ (Fil‘𝐴) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
97, 8syl 17 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
10 restsspw 17477 . . . . . 6 (𝐹t 𝐴) ⊆ 𝒫 𝐴
114sspwd 4617 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
1210, 11sstrid 4006 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝒫 𝑋)
13 filtop 23878 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
1413adantr 480 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝑋𝐹)
15 fbasweak 23888 . . . . 5 (((𝐹t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹t 𝐴) ⊆ 𝒫 𝑋𝑋𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝑋))
169, 12, 14, 15syl3anc 1370 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝑋))
171adantr 480 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐹 ∈ (fBas‘𝑋))
18 trfilss 23912 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)
19 fgss 23896 . . . 4 (((𝐹t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ (𝑋filGen𝐹))
2016, 17, 18, 19syl3anc 1370 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ (𝑋filGen𝐹))
21 fgfil 23898 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
2221adantr 480 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen𝐹) = 𝐹)
2320, 22sseqtrd 4035 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ 𝐹)
24 filelss 23875 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
2524ex 412 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹𝑥𝑋))
2625adantr 480 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹𝑥𝑋))
27 elrestr 17474 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝑥𝐹) → (𝑥𝐴) ∈ (𝐹t 𝐴))
28273expa 1117 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → (𝑥𝐴) ∈ (𝐹t 𝐴))
29 inss1 4244 . . . . . . 7 (𝑥𝐴) ⊆ 𝑥
30 sseq1 4020 . . . . . . . 8 (𝑦 = (𝑥𝐴) → (𝑦𝑥 ↔ (𝑥𝐴) ⊆ 𝑥))
3130rspcev 3621 . . . . . . 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 23894 . . . . 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 4000 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹t 𝐴)))
3923, 38eqssd 4012 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 1536  wcel 2105  wrex 3067  cdif 3959  cin 3961  wss 3962  𝒫 cpw 4604  cfv 6562  (class class class)co 7430  t crest 17466  fBascfbas 21369  filGencfg 21370  Filcfil 23868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-rest 17468  df-fbas 21378  df-fg 21379  df-fil 23869
This theorem is referenced by:  cfilres  25343
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