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Theorem lpcls 21390
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 21356 . . . . . . 7 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 lpcls.1 . . . . . . . . . 10 𝑋 = 𝐽
32clsss3 21085 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
43ssdifssd 3958 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
52clsss3 21085 . . . . . . . 8 ((𝐽 ∈ Top ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
64, 5syldan 581 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
71, 6sylan 571 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
87sseld 3808 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥𝑋))
9 ssdifss 3951 . . . . . . . . . . 11 (𝑆𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
102clscld 21073 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
111, 9, 10syl2an 585 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
1211adantr 468 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
132t1sncld 21352 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
1413adantlr 697 . . . . . . . . . . . 12 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
15 uncld 21067 . . . . . . . . . . . 12 (({𝑥} ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽)) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
1614, 12, 15syl2anc 575 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
172sscls 21082 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
181, 9, 17syl2an 585 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
19 ssundif 4259 . . . . . . . . . . . . 13 (𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2018, 19sylibr 225 . . . . . . . . . . . 12 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2120adantr 468 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
222clsss2 21098 . . . . . . . . . . 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 575 . . . . . . . . . 10 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
24 ssundif 4259 . . . . . . . . . 10 (((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2523, 24sylib 209 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
262clsss2 21098 . . . . . . . . 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 575 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2827sseld 3808 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2928ex 399 . . . . . 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 468 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝐽 ∈ Top)
331, 3sylan 571 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
3433ssdifssd 3958 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
352sscls 21082 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
361, 35sylan 571 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
3736ssdifd 3956 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑆 ∖ {𝑥}) ⊆ (((cls‘𝐽)‘𝑆) ∖ {𝑥}))
382clsss 21080 . . . . . 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 1483 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})))
4039sseld 3808 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
4131, 40impbid 203 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
422islp 21166 . . . . 5 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
433, 42syldan 581 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
441, 43sylan 571 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
452islp 21166 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
461, 45sylan 571 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
4741, 44, 463bitr4d 302 . 2 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
4847eqrdv 2815 1 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  cdif 3777  cun 3778  wss 3780  {csn 4381   cuni 4641  cfv 6108  Topctop 20919  Clsdccld 21042  clsccl 21044  limPtclp 21160  Frect1 21333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-iin 4726  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5230  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-top 20920  df-cld 21045  df-cls 21047  df-lp 21162  df-t1 21340
This theorem is referenced by:  perfcls  21391
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