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Theorem trnei 23740
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 23735 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 22759 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐽 ∈ Top)
213ad2ant1 1130 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐽 ∈ Top)
3 simp2 1134 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐴 βŠ† π‘Œ)
4 toponuni 22760 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐽)
543ad2ant1 1130 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ π‘Œ = βˆͺ 𝐽)
63, 5sseqtrd 4015 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐴 βŠ† βˆͺ 𝐽)
7 simp3 1135 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝑃 ∈ π‘Œ)
87, 5eleqtrd 2827 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝑃 ∈ βˆͺ 𝐽)
9 eqid 2724 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
109neindisj2 22971 . . 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 1368 . 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 1133 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘Œ))
137snssd 4805 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ {𝑃} βŠ† π‘Œ)
14 snnzg 4771 . . . . 5 (𝑃 ∈ π‘Œ β†’ {𝑃} β‰  βˆ…)
15143ad2ant3 1132 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ {𝑃} β‰  βˆ…)
16 neifil 23728 . . . 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 1368 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 βŠ† π‘Œ ∧ 𝑃 ∈ π‘Œ) β†’ ((neiβ€˜π½)β€˜{𝑃}) ∈ (Filβ€˜π‘Œ))
18 trfil2 23735 . . 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 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4315  {csn 4621  βˆͺ cuni 4900  β€˜cfv 6534  (class class class)co 7402   β†Ύt crest 17371  Topctop 22739  TopOnctopon 22756  clsccl 22866  neicnei 22945  Filcfil 23693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-rest 17373  df-fbas 21231  df-top 22740  df-topon 22757  df-cld 22867  df-ntr 22868  df-cls 22869  df-nei 22946  df-fil 23694
This theorem is referenced by:  flfcntr  23891  cnextfun  23912  cnextfvval  23913  cnextf  23914  cnextcn  23915  cnextfres1  23916  cnextucn  24152  ucnextcn  24153  limcflflem  25753  rrhre  33520
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