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Theorem trnei 23795
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 23790 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 22814 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐽 ∈ Top)
213ad2ant1 1131 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐽 ∈ Top)
3 simp2 1135 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐴 βŠ† π‘Œ)
4 toponuni 22815 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐽)
543ad2ant1 1131 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ π‘Œ = βˆͺ 𝐽)
63, 5sseqtrd 4020 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐴 βŠ† βˆͺ 𝐽)
7 simp3 1136 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝑃 ∈ π‘Œ)
87, 5eleqtrd 2831 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝑃 ∈ βˆͺ 𝐽)
9 eqid 2728 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
109neindisj2 23026 . . 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 1369 . 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 1134 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘Œ))
137snssd 4813 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ {𝑃} βŠ† π‘Œ)
14 snnzg 4779 . . . . 5 (𝑃 ∈ π‘Œ β†’ {𝑃} β‰  βˆ…)
15143ad2ant3 1133 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ {𝑃} β‰  βˆ…)
16 neifil 23783 . . . 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 1369 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ ((neiβ€˜π½)β€˜{𝑃}) ∈ (Filβ€˜π‘Œ))
18 trfil2 23790 . . 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 583 . 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 282 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 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  βˆ€wral 3058   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4908  β€˜cfv 6548  (class class class)co 7420   β†Ύt crest 17401  Topctop 22794  TopOnctopon 22811  clsccl 22921  neicnei 23000  Filcfil 23748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-rest 17403  df-fbas 21275  df-top 22795  df-topon 22812  df-cld 22922  df-ntr 22923  df-cls 22924  df-nei 23001  df-fil 23749
This theorem is referenced by:  flfcntr  23946  cnextfun  23967  cnextfvval  23968  cnextf  23969  cnextcn  23970  cnextfres1  23971  cnextucn  24207  ucnextcn  24208  limcflflem  25808  rrhre  33622
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