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

Theorem trnei 23916
Description: The trace, over a set 𝐴, of the filter of the neighborhoods of a point 𝑃 is a filter iff 𝑃 belongs to the closure of 𝐴. (This is trfil2 23911 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
trnei ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴)))

Proof of Theorem trnei
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 topontop 22935 . . . 4 (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top)
213ad2ant1 1132 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ Top)
3 simp2 1136 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴𝑌)
4 toponuni 22936 . . . . 5 (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = 𝐽)
543ad2ant1 1132 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑌 = 𝐽)
63, 5sseqtrd 4036 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴 𝐽)
7 simp3 1137 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃𝑌)
87, 5eleqtrd 2841 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃 𝐽)
9 eqid 2735 . . . 4 𝐽 = 𝐽
109neindisj2 23147 . . 3 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑃 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
112, 6, 8, 10syl3anc 1370 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
12 simp1 1135 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ (TopOn‘𝑌))
137snssd 4814 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ⊆ 𝑌)
14 snnzg 4779 . . . . 5 (𝑃𝑌 → {𝑃} ≠ ∅)
15143ad2ant3 1134 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ≠ ∅)
16 neifil 23904 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
1712, 13, 15, 16syl3anc 1370 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18 trfil2 23911 . . 3 ((((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
1917, 3, 18syl2anc 584 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
2011, 19bitr4d 282 1 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  cin 3962  wss 3963  c0 4339  {csn 4631   cuni 4912  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  TopOnctopon 22932  clsccl 23042  neicnei 23121  Filcfil 23869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-rest 17469  df-fbas 21379  df-top 22916  df-topon 22933  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-fil 23870
This theorem is referenced by:  flfcntr  24067  cnextfun  24088  cnextfvval  24089  cnextf  24090  cnextcn  24091  cnextfres1  24092  cnextucn  24328  ucnextcn  24329  limcflflem  25930  rrhre  33984
  Copyright terms: Public domain W3C validator