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Theorem lpcls 22837
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 22803 . . . . . . 7 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 lpcls.1 . . . . . . . . . 10 𝑋 = 𝐽
32clsss3 22532 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
43ssdifssd 4140 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
52clsss3 22532 . . . . . . . 8 ((𝐽 ∈ Top ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
64, 5syldan 592 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
71, 6sylan 581 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
87sseld 3979 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥𝑋))
9 ssdifss 4133 . . . . . . . . . . 11 (𝑆𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
102clscld 22520 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
111, 9, 10syl2an 597 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
1211adantr 482 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
132t1sncld 22799 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
1413adantlr 714 . . . . . . . . . . . 12 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
15 uncld 22514 . . . . . . . . . . . 12 (({𝑥} ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽)) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
1614, 12, 15syl2anc 585 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
172sscls 22529 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
181, 9, 17syl2an 597 . . . . . . . . . . . . 13 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
19 ssundif 4483 . . . . . . . . . . . . 13 (𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2018, 19sylibr 233 . . . . . . . . . . . 12 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2120adantr 482 . . . . . . . . . . 11 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
222clsss2 22545 . . . . . . . . . . 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 585 . . . . . . . . . 10 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
24 ssundif 4483 . . . . . . . . . 10 (((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2523, 24sylib 217 . . . . . . . . 9 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
262clsss2 22545 . . . . . . . . 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 585 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
2827sseld 3979 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝑆𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2928ex 414 . . . . . 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 482 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝐽 ∈ Top)
331, 3sylan 581 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
3433ssdifssd 4140 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
352sscls 22529 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
361, 35sylan 581 . . . . . . 7 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
3736ssdifd 4138 . . . . . 6 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑆 ∖ {𝑥}) ⊆ (((cls‘𝐽)‘𝑆) ∖ {𝑥}))
382clsss 22527 . . . . . 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 1372 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})))
4039sseld 3979 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
4131, 40impbid 211 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
422islp 22613 . . . . 5 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
433, 42syldan 592 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
441, 43sylan 581 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
452islp 22613 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
461, 45sylan 581 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
4741, 44, 463bitr4d 311 . 2 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
4847eqrdv 2731 1 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cdif 3943  cun 3944  wss 3946  {csn 4624   cuni 4904  cfv 6535  Topctop 22364  Clsdccld 22489  clsccl 22491  limPtclp 22607  Frect1 22780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-top 22365  df-cld 22492  df-cls 22494  df-lp 22609  df-t1 22787
This theorem is referenced by:  perfcls  22838
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