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Theorem clslp 21174
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 21143 . . . . . . . . . . . 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 4473 . . . . . . . . . . . . 13 𝑥𝑆 → (𝑆 ∖ {𝑥}) = 𝑆)
65ineq2d 3957 . . . . . . . . . . . 12 𝑥𝑆 → (𝑛 ∩ (𝑆 ∖ {𝑥})) = (𝑛𝑆))
76neeq1d 2991 . . . . . . . . . . 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 3106 . . . . . . 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 21085 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
1514sselda 3744 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥𝑋)
161islp2 21171 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑥𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
1712, 13, 15, 16syl3anc 1477 . . . . . . 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 3896 . . . . 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 3750 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
241sscls 21082 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
251lpsscls 21167 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆))
2624, 25unssd 3932 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
2723, 26eqssd 3761 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 1632  wcel 2139  wne 2932  wral 3050  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058  {csn 4321   cuni 4588  cfv 6049  Topctop 20920  clsccl 21044  neicnei 21123  limPtclp 21160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20921  df-cld 21045  df-ntr 21046  df-cls 21047  df-nei 21124  df-lp 21162
This theorem is referenced by:  islpi  21175  cldlp  21176  perfcls  21391
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