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Theorem trnei 23387
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 23382 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 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π΄) ↔ (((neiβ€˜π½)β€˜{𝑃}) β†Ύt 𝐴) ∈ (Filβ€˜π΄)))
Proof of Theorem trnei
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 topontop 22406 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐽 ∈ Top)
213ad2ant1 1133 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐽 ∈ Top)
3 simp2 1137 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐴 βŠ† π‘Œ)
4 toponuni 22407 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐽)
543ad2ant1 1133 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ π‘Œ = βˆͺ 𝐽)
63, 5sseqtrd 4021 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐴 βŠ† βˆͺ 𝐽)
7 simp3 1138 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝑃 ∈ π‘Œ)
87, 5eleqtrd 2835 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝑃 ∈ βˆͺ 𝐽)
9 eqid 2732 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
109neindisj2 22618 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† βˆͺ 𝐽 ∧ 𝑃 ∈ βˆͺ 𝐽) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π΄) ↔ βˆ€π‘£ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑣 ∩ 𝐴) β‰  βˆ…))
112, 6, 8, 10syl3anc 1371 . 2 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π΄) ↔ βˆ€π‘£ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑣 ∩ 𝐴) β‰  βˆ…))
12 simp1 1136 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘Œ))
137snssd 4811 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ {𝑃} βŠ† π‘Œ)
14 snnzg 4777 . . . . 5 (𝑃 ∈ π‘Œ β†’ {𝑃} β‰  βˆ…)
15143ad2ant3 1135 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ {𝑃} β‰  βˆ…)
16 neifil 23375 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ {𝑃} βŠ† π‘Œ ∧ {𝑃} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝑃}) ∈ (Filβ€˜π‘Œ))
1712, 13, 15, 16syl3anc 1371 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ ((neiβ€˜π½)β€˜{𝑃}) ∈ (Filβ€˜π‘Œ))
18 trfil2 23382 . . 3 ((((neiβ€˜π½)β€˜{𝑃}) ∈ (Filβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ) β†’ ((((neiβ€˜π½)β€˜{𝑃}) β†Ύt 𝐴) ∈ (Filβ€˜π΄) ↔ βˆ€π‘£ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑣 ∩ 𝐴) β‰  βˆ…))
1917, 3, 18syl2anc 584 . 2 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ ((((neiβ€˜π½)β€˜{𝑃}) β†Ύt 𝐴) ∈ (Filβ€˜π΄) ↔ βˆ€π‘£ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑣 ∩ 𝐴) β‰  βˆ…))
2011, 19bitr4d 281 1 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π΄) ↔ (((neiβ€˜π½)β€˜{𝑃}) β†Ύt 𝐴) ∈ (Filβ€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  {csn 4627  βˆͺ cuni 4907  β€˜cfv 6540  (class class class)co 7405   β†Ύt crest 17362  Topctop 22386  TopOnctopon 22403  clsccl 22513  neicnei 22592  Filcfil 23340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-rest 17364  df-fbas 20933  df-top 22387  df-topon 22404  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-fil 23341
This theorem is referenced by:  flfcntr  23538  cnextfun  23559  cnextfvval  23560  cnextf  23561  cnextcn  23562  cnextfres1  23563  cnextucn  23799  ucnextcn  23800  limcflflem  25388  rrhre  32989
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