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Theorem trnei 21601
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 21596 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 20636 . . . 4 (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top)
213ad2ant1 1080 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ Top)
3 simp2 1060 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴𝑌)
4 toponuni 20637 . . . . 5 (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = 𝐽)
543ad2ant1 1080 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑌 = 𝐽)
63, 5sseqtrd 3625 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴 𝐽)
7 simp3 1061 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃𝑌)
87, 5eleqtrd 2706 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃 𝐽)
9 eqid 2626 . . . 4 𝐽 = 𝐽
109neindisj2 20832 . . 3 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑃 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
112, 6, 8, 10syl3anc 1323 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
12 simp1 1059 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ (TopOn‘𝑌))
137snssd 4314 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ⊆ 𝑌)
14 snnzg 4283 . . . . 5 (𝑃𝑌 → {𝑃} ≠ ∅)
15143ad2ant3 1082 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ≠ ∅)
16 neifil 21589 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
1712, 13, 15, 16syl3anc 1323 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18 trfil2 21596 . . 3 ((((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
1917, 3, 18syl2anc 692 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
2011, 19bitr4d 271 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 196  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  cin 3559  wss 3560  c0 3896  {csn 4153   cuni 4407  cfv 5850  (class class class)co 6605  t crest 15997  Topctop 20612  TopOnctopon 20613  clsccl 20727  neicnei 20806  Filcfil 21554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-rest 15999  df-fbas 19657  df-top 20616  df-topon 20618  df-cld 20728  df-ntr 20729  df-cls 20730  df-nei 20807  df-fil 21555
This theorem is referenced by:  flfcntr  21752  cnextfun  21773  cnextfvval  21774  cnextf  21775  cnextcn  21776  cnextfres1  21777  cnextucn  22012  ucnextcn  22013  limcflflem  23545  rrhre  29839
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