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Theorem lpcls 22261
Description: The limit points of the closure of a subset are the same as the limit points of the set in a T1 space. (Contributed by Mario Carneiro, 26-Dec-2016.)
Hypothesis
Ref Expression
lpcls.1 𝑋 = 𝐽
Assertion
Ref Expression
lpcls ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))

Proof of Theorem lpcls
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 t1top 22227 . . . . . . 7 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 lpcls.1 . . . . . . . . . 10 𝑋 = 𝐽
32clsss3 21956 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
43ssdifssd 4057 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
52clsss3 21956 . . . . . . . 8 ((𝐽 ∈ Top ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
64, 5syldan 594 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
71, 6sylan 583 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
87sseld 3900 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥𝑋))
9 ssdifss 4050 . . . . . . . . . . 11 (𝑆𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
102clscld 21944 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
111, 9, 10syl2an 599 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
1211adantr 484 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
132t1sncld 22223 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
1413adantlr 715 . . . . . . . . . . . 12 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
15 uncld 21938 . . . . . . . . . . . 12 (({𝑥} ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽)) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
1614, 12, 15syl2anc 587 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
172sscls 21953 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
181, 9, 17syl2an 599 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
19 ssundif 4399 . . . . . . . . . . . . 13 (𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2018, 19sylibr 237 . . . . . . . . . . . 12 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2120adantr 484 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
222clsss2 21969 . . . . . . . . . . 11 ((({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))) → ((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2316, 21, 22syl2anc 587 . . . . . . . . . 10 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
24 ssundif 4399 . . . . . . . . . 10 (((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2523, 24sylib 221 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
262clsss2 21969 . . . . . . . . 9 ((((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽) ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2712, 25, 26syl2anc 587 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2827sseld 3900 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2928ex 416 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥𝑋 → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))))
3029com23 86 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → (𝑥𝑋𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))))
318, 30mpdd 43 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
321adantr 484 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝐽 ∈ Top)
331, 3sylan 583 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
3433ssdifssd 4057 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
352sscls 21953 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
361, 35sylan 583 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
3736ssdifd 4055 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑆 ∖ {𝑥}) ⊆ (((cls‘𝐽)‘𝑆) ∖ {𝑥}))
382clsss 21951 . . . . . 6 ((𝐽 ∈ Top ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ (((cls‘𝐽)‘𝑆) ∖ {𝑥})) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})))
3932, 34, 37, 38syl3anc 1373 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})))
4039sseld 3900 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
4131, 40impbid 215 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
422islp 22037 . . . . 5 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
433, 42syldan 594 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
441, 43sylan 583 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
452islp 22037 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
461, 45sylan 583 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
4741, 44, 463bitr4d 314 . 2 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
4847eqrdv 2735 1 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  cdif 3863  cun 3864  wss 3866  {csn 4541   cuni 4819  cfv 6380  Topctop 21790  Clsdccld 21913  clsccl 21915  limPtclp 22031  Frect1 22204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-top 21791  df-cld 21916  df-cls 21918  df-lp 22033  df-t1 22211
This theorem is referenced by:  perfcls  22262
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