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Theorem trnei 21506
 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 21501 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 20541 . . . 4 (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top)
213ad2ant1 1075 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ Top)
3 simp2 1055 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴𝑌)
4 toponuni 20542 . . . . 5 (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = 𝐽)
543ad2ant1 1075 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑌 = 𝐽)
63, 5sseqtrd 3604 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐴 𝐽)
7 simp3 1056 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃𝑌)
87, 5eleqtrd 2690 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝑃 𝐽)
9 eqid 2610 . . . 4 𝐽 = 𝐽
109neindisj2 20737 . . 3 ((𝐽 ∈ Top ∧ 𝐴 𝐽𝑃 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
112, 6, 8, 10syl3anc 1318 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
12 simp1 1054 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → 𝐽 ∈ (TopOn‘𝑌))
137snssd 4281 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ⊆ 𝑌)
14 snnzg 4251 . . . . 5 (𝑃𝑌 → {𝑃} ≠ ∅)
15143ad2ant3 1077 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → {𝑃} ≠ ∅)
16 neifil 21494 . . . 4 ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
1712, 13, 15, 16syl3anc 1318 . . 3 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18 trfil2 21501 . . 3 ((((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
1917, 3, 18syl2anc 691 . 2 ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣𝐴) ≠ ∅))
2011, 19bitr4d 270 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 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  TopOnctopon 20518  clsccl 20632  neicnei 20711  Filcfil 21459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rest 15906  df-fbas 19564  df-top 20521  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-fil 21460 This theorem is referenced by:  flfcntr  21657  cnextfun  21678  cnextfvval  21679  cnextf  21680  cnextcn  21681  cnextfres1  21682  cnextucn  21917  ucnextcn  21918  limcflflem  23450  rrhre  29393
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