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Theorem trnei 22104
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 22099 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 21125 . . . 4 (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top)
213ad2ant1 1124 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ Top)
3 simp2 1128 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴𝑌)
4 toponuni 21126 . . . . 5 (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = 𝐽)
543ad2ant1 1124 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑌 = 𝐽)
63, 5sseqtrd 3860 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴 𝐽)
7 simp3 1129 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃𝑌)
87, 5eleqtrd 2861 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃 𝐽)
9 eqid 2778 . . . 4 𝐽 = 𝐽
109neindisj2 21335 . . 3 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑃 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
112, 6, 8, 10syl3anc 1439 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
12 simp1 1127 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ (TopOn‘𝑌))
137snssd 4571 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ⊆ 𝑌)
14 snnzg 4541 . . . . 5 (𝑃𝑌 → {𝑃} ≠ ∅)
15143ad2ant3 1126 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ≠ ∅)
16 neifil 22092 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
1712, 13, 15, 16syl3anc 1439 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18 trfil2 22099 . . 3 ((((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
1917, 3, 18syl2anc 579 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
2011, 19bitr4d 274 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 198  w3a 1071   = wceq 1601  wcel 2107  wne 2969  wral 3090  cin 3791  wss 3792  c0 4141  {csn 4398   cuni 4671  cfv 6135  (class class class)co 6922  t crest 16467  Topctop 21105  TopOnctopon 21122  clsccl 21230  neicnei 21309  Filcfil 22057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-rest 16469  df-fbas 20139  df-top 21106  df-topon 21123  df-cld 21231  df-ntr 21232  df-cls 21233  df-nei 21310  df-fil 22058
This theorem is referenced by:  flfcntr  22255  cnextfun  22276  cnextfvval  22277  cnextf  22278  cnextcn  22279  cnextfres1  22280  cnextucn  22515  ucnextcn  22516  limcflflem  24081  rrhre  30663
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