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Theorem clslp 22651
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 22620 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†) ∧ 𝑛 ∈ ((neiβ€˜π½)β€˜{π‘₯}))) β†’ (𝑛 ∩ 𝑆) β‰  βˆ…)
32expr 457 . . . . . . . . . . 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 481 . . . . . . . . . 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 4801 . . . . . . . . . . . . 13 (Β¬ π‘₯ ∈ 𝑆 β†’ (𝑆 βˆ– {π‘₯}) = 𝑆)
65ineq2d 4212 . . . . . . . . . . . 12 (Β¬ π‘₯ ∈ 𝑆 β†’ (𝑛 ∩ (𝑆 βˆ– {π‘₯})) = (𝑛 ∩ 𝑆))
76neeq1d 3000 . . . . . . . . . . 11 (Β¬ π‘₯ ∈ 𝑆 β†’ ((𝑛 ∩ (𝑆 βˆ– {π‘₯})) β‰  βˆ… ↔ (𝑛 ∩ 𝑆) β‰  βˆ…))
87adantl 482 . . . . . . . . . 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 258 . . . . . . . . 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 413 . . . . . . . 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 3152 . . . . . . 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 765 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ Top)
13 simplr 767 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
141clsss3 22562 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
1514sselda 3982 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ π‘₯ ∈ 𝑋)
161islp2 22648 . . . . . . . 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 1371 . . . . . . 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 258 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (Β¬ π‘₯ ∈ 𝑆 β†’ π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†)))
1918orrd 861 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∈ 𝑆 ∨ π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†)))
20 elun 4148 . . . . 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 413 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜π‘†) β†’ π‘₯ ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†))))
2322ssrdv 3988 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
241sscls 22559 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
251lpsscls 22644 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((limPtβ€˜π½)β€˜π‘†) βŠ† ((clsβ€˜π½)β€˜π‘†))
2624, 25unssd 4186 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) βŠ† ((clsβ€˜π½)β€˜π‘†))
2723, 26eqssd 3999 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   βˆ– cdif 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22394  clsccl 22521  neicnei 22600  limPtclp 22637
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22395  df-cld 22522  df-ntr 22523  df-cls 22524  df-nei 22601  df-lp 22639
This theorem is referenced by:  islpi  22652  cldlp  22653  perfcls  22868
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