MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fgtr Structured version   Visualization version   GIF version

Theorem fgtr 22493
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 22451 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbncp 22442 . . . . . . . 8 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ (𝑋𝐴) ∈ 𝐹)
31, 2sylan 583 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ¬ (𝑋𝐴) ∈ 𝐹)
4 filelss 22455 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
5 trfil3 22491 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋𝐴) ∈ 𝐹))
64, 5syldan 594 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → ((𝐹t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑋𝐴) ∈ 𝐹))
73, 6mpbird 260 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (Fil‘𝐴))
8 filfbas 22451 . . . . . 6 ((𝐹t 𝐴) ∈ (Fil‘𝐴) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
97, 8syl 17 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
10 restsspw 16703 . . . . . 6 (𝐹t 𝐴) ⊆ 𝒫 𝐴
114sspwd 4537 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
1210, 11sstrid 3964 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝒫 𝑋)
13 filtop 22458 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
1413adantr 484 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝑋𝐹)
15 fbasweak 22468 . . . . 5 (((𝐹t 𝐴) ∈ (fBas‘𝐴) ∧ (𝐹t 𝐴) ⊆ 𝒫 𝑋𝑋𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝑋))
169, 12, 14, 15syl3anc 1368 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ∈ (fBas‘𝑋))
171adantr 484 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐹 ∈ (fBas‘𝑋))
18 trfilss 22492 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)
19 fgss 22476 . . . 4 (((𝐹t 𝐴) ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (fBas‘𝑋) ∧ (𝐹t 𝐴) ⊆ 𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ (𝑋filGen𝐹))
2016, 17, 18, 19syl3anc 1368 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ (𝑋filGen𝐹))
21 fgfil 22478 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
2221adantr 484 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen𝐹) = 𝐹)
2320, 22sseqtrd 3993 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) ⊆ 𝐹)
24 filelss 22455 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
2524ex 416 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹𝑥𝑋))
2625adantr 484 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹𝑥𝑋))
27 elrestr 16700 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝑥𝐹) → (𝑥𝐴) ∈ (𝐹t 𝐴))
28273expa 1115 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → (𝑥𝐴) ∈ (𝐹t 𝐴))
29 inss1 4190 . . . . . . 7 (𝑥𝐴) ⊆ 𝑥
30 sseq1 3978 . . . . . . . 8 (𝑦 = (𝑥𝐴) → (𝑦𝑥 ↔ (𝑥𝐴) ⊆ 𝑥))
3130rspcev 3609 . . . . . . 7 (((𝑥𝐴) ∈ (𝐹t 𝐴) ∧ (𝑥𝐴) ⊆ 𝑥) → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)
3228, 29, 31sylancl 589 . . . . . 6 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) ∧ 𝑥𝐹) → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)
3332ex 416 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹 → ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥))
3426, 33jcad 516 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹 → (𝑥𝑋 ∧ ∃𝑦 ∈ (𝐹t 𝐴)𝑦𝑥)))
35 elfg 22474 . . . . 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 262 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑥𝐹𝑥 ∈ (𝑋filGen(𝐹t 𝐴))))
3837ssrdv 3959 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐹 ⊆ (𝑋filGen(𝐹t 𝐴)))
3923, 38eqssd 3970 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wrex 3134  cdif 3916  cin 3918  wss 3919  𝒫 cpw 4522  cfv 6344  (class class class)co 7146  t crest 16692  fBascfbas 20528  filGencfg 20529  Filcfil 22448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-rest 16694  df-fbas 20537  df-fg 20538  df-fil 22449
This theorem is referenced by:  cfilres  23898
  Copyright terms: Public domain W3C validator