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Theorem clslp 21075
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 = 𝐽
Assertion
Ref Expression
clslp ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))

Proof of Theorem clslp
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
21neindisj 21044 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑥}))) → (𝑛𝑆) ≠ ∅)
32expr 644 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛𝑆) ≠ ∅))
43adantr 472 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛𝑆) ≠ ∅))
5 difsn 4436 . . . . . . . . . . . . 13 𝑥𝑆 → (𝑆 ∖ {𝑥}) = 𝑆)
65ineq2d 3922 . . . . . . . . . . . 12 𝑥𝑆 → (𝑛 ∩ (𝑆 ∖ {𝑥})) = (𝑛𝑆))
76neeq1d 2955 . . . . . . . . . . 11 𝑥𝑆 → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛𝑆) ≠ ∅))
87adantl 473 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥𝑆) → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛𝑆) ≠ ∅))
94, 8sylibrd 249 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
109ex 449 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥𝑆 → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)))
1110ralrimdv 3070 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥𝑆 → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
12 simpll 807 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
13 simplr 809 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝑋)
141clsss3 20986 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
1514sselda 3709 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥𝑋)
161islp2 21072 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑥𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
1712, 13, 15, 16syl3anc 1439 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
1811, 17sylibrd 249 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥𝑆𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
1918orrd 392 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥𝑆𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
20 elun 3861 . . . . 5 (𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑥𝑆𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
2119, 20sylibr 224 . . . 4 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
2221ex 449 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
2322ssrdv 3715 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
241sscls 20983 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
251lpsscls 21068 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆))
2624, 25unssd 3897 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
2723, 26eqssd 3726 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1596  wcel 2103  wne 2896  wral 3014  cdif 3677  cun 3678  cin 3679  wss 3680  c0 4023  {csn 4285   cuni 4544  cfv 6001  Topctop 20821  clsccl 20945  neicnei 21024  limPtclp 21061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-top 20822  df-cld 20946  df-ntr 20947  df-cls 20948  df-nei 21025  df-lp 21063
This theorem is referenced by:  islpi  21076  cldlp  21077  perfcls  21292
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