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Theorem perfcls 21079
Description: A subset of a perfect space is perfect iff its closure is perfect (and the closure is an actual perfect set, since it is both closed and perfect in the subspace topology). (Contributed by Mario Carneiro, 26-Dec-2016.)
Hypothesis
Ref Expression
lpcls.1 𝑋 = 𝐽
Assertion
Ref Expression
perfcls ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Perf ↔ (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Perf))

Proof of Theorem perfcls
StepHypRef Expression
1 lpcls.1 . . . . 5 𝑋 = 𝐽
21lpcls 21078 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
32sseq2d 3612 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆)))
4 t1top 21044 . . . . . 6 (𝐽 ∈ Fre → 𝐽 ∈ Top)
51clslp 20862 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
64, 5sylan 488 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
76sseq1d 3611 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆)))
8 ssequn1 3761 . . . . 5 (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
9 ssun2 3755 . . . . . 6 ((limPt‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))
10 eqss 3598 . . . . . 6 ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆) ↔ ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆) ∧ ((limPt‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
119, 10mpbiran2 953 . . . . 5 ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆))
128, 11bitri 264 . . . 4 (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆))
137, 12syl6bbr 278 . . 3 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
143, 13bitr2d 269 . 2 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
15 eqid 2621 . . . 4 (𝐽t 𝑆) = (𝐽t 𝑆)
161, 15restperf 20898 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Perf ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
174, 16sylan 488 . 2 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Perf ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
181clsss3 20773 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
19 eqid 2621 . . . . 5 (𝐽t ((cls‘𝐽)‘𝑆)) = (𝐽t ((cls‘𝐽)‘𝑆))
201, 19restperf 20898 . . . 4 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → ((𝐽t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
2118, 20syldan 487 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
224, 21sylan 488 . 2 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((𝐽t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
2314, 17, 223bitr4d 300 1 ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Perf ↔ (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Perf))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cun 3553  wss 3555   cuni 4402  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617  clsccl 20732  limPtclp 20848  Perfcperf 20849  Frect1 21021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-t1 21028
This theorem is referenced by: (None)
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