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Theorem clslp 22652
Description: The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
clslp ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
Proof of Theorem clslp
Dummy variables 𝑛 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . . . . . . . . . . 13 𝑋 = βˆͺ 𝐽
21neindisj 22621 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†) ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{π‘₯}))) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)
32expr 458 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…))
43adantr 482 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ Β¬ π‘₯ ∈ 𝑆) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…))
5 difsn 4802 . . . . . . . . . . . . 13 (Β¬ π‘₯ ∈ 𝑆 β†’ (𝑆 βˆ– {π‘₯}) = 𝑆)
65ineq2d 4213 . . . . . . . . . . . 12 (Β¬ π‘₯ ∈ 𝑆 β†’ (𝑛 ∩ (𝑆 βˆ– {π‘₯})) = (𝑛 ∩ 𝑆))
76neeq1d 3001 . . . . . . . . . . 11 (Β¬ π‘₯ ∈ 𝑆 β†’ ((𝑛 ∩ (𝑆 βˆ– {π‘₯})) β‰  βˆ… ↔ (𝑛 ∩ 𝑆) β‰  βˆ…))
87adantl 483 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ Β¬ π‘₯ ∈ 𝑆) β†’ ((𝑛 ∩ (𝑆 βˆ– {π‘₯})) β‰  βˆ… ↔ (𝑛 ∩ 𝑆) β‰  βˆ…))
94, 8sylibrd 259 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ Β¬ π‘₯ ∈ 𝑆) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ (𝑛 ∩ (𝑆 βˆ– {π‘₯})) β‰  βˆ…))
109ex 414 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (Β¬ π‘₯ ∈ 𝑆 β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ (𝑛 ∩ (𝑆 βˆ– {π‘₯})) β‰  βˆ…)))
1110ralrimdv 3153 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (Β¬ π‘₯ ∈ 𝑆 β†’ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{π‘₯})(𝑛 ∩ (𝑆 βˆ– {π‘₯})) β‰  βˆ…))
12 simpll 766 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ Top)
13 simplr 768 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
141clsss3 22563 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
1514sselda 3983 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ π‘₯ ∈ 𝑋)
161islp2 22649 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{π‘₯})(𝑛 ∩ (𝑆 βˆ– {π‘₯})) β‰  βˆ…))
1712, 13, 15, 16syl3anc 1372 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{π‘₯})(𝑛 ∩ (𝑆 βˆ– {π‘₯})) β‰  βˆ…))
1811, 17sylibrd 259 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (Β¬ π‘₯ ∈ 𝑆 β†’ π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†)))
1918orrd 862 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∈ 𝑆 ∨ π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†)))
20 elun 4149 . . . . 5 (π‘₯ ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) ↔ (π‘₯ ∈ 𝑆 ∨ π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†)))
2119, 20sylibr 233 . . . 4 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ π‘₯ ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
2221ex 414 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†) β†’ π‘₯ ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†))))
2322ssrdv 3989 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
241sscls 22560 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
251lpsscls 22645 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((limPtβ€˜π½)β€˜π‘†) βŠ† ((clsβ€˜π½)β€˜π‘†))
2624, 25unssd 4187 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) βŠ† ((clsβ€˜π½)β€˜π‘†))
2723, 26eqssd 4000 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   βˆ– cdif 3946   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  clsccl 22522  neicnei 22601  limPtclp 22638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-lp 22640
This theorem is referenced by:  islpi  22653  cldlp  22654  perfcls  22869
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