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Theorem trnei 23901
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 23896 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 22920 . . . 4 (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top)
213ad2ant1 1133 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ Top)
3 simp2 1137 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴𝑌)
4 toponuni 22921 . . . . 5 (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = 𝐽)
543ad2ant1 1133 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑌 = 𝐽)
63, 5sseqtrd 4019 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴 𝐽)
7 simp3 1138 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃𝑌)
87, 5eleqtrd 2842 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃 𝐽)
9 eqid 2736 . . . 4 𝐽 = 𝐽
109neindisj2 23132 . . 3 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑃 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
112, 6, 8, 10syl3anc 1372 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
12 simp1 1136 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ (TopOn‘𝑌))
137snssd 4808 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ⊆ 𝑌)
14 snnzg 4773 . . . . 5 (𝑃𝑌 → {𝑃} ≠ ∅)
15143ad2ant3 1135 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ≠ ∅)
16 neifil 23889 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
1712, 13, 15, 16syl3anc 1372 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18 trfil2 23896 . . 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 1539  wcel 2107  wne 2939  wral 3060  cin 3949  wss 3950  c0 4332  {csn 4625   cuni 4906  cfv 6560  (class class class)co 7432  t crest 17466  Topctop 22900  TopOnctopon 22917  clsccl 23027  neicnei 23106  Filcfil 23854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-rest 17468  df-fbas 21362  df-top 22901  df-topon 22918  df-cld 23028  df-ntr 23029  df-cls 23030  df-nei 23107  df-fil 23855
This theorem is referenced by:  flfcntr  24052  cnextfun  24073  cnextfvval  24074  cnextf  24075  cnextcn  24076  cnextfres1  24077  cnextucn  24313  ucnextcn  24314  limcflflem  25916  rrhre  34023
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