Theorem perfcls 22079

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 ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf))

Proof of Theorem perfcls

Step	Hyp	Ref	Expression
1		lpcls.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	lpcls 22078	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
3	2	sseq2d 3926	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆)))
4		t1top 22044	. . . . . 6 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
5	1	clslp 21862	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
6	4, 5	sylan 583	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
7	6	sseq1d 3925	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆)))
8		ssequn1 4087	. . . . 5 ⊢ (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
9		ssun2 4080	. . . . . 6 ⊢ ((limPt‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))
10		eqss 3909	. . . . . 6 ⊢ ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆) ↔ ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆) ∧ ((limPt‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
11	9, 10	mpbiran2 709	. . . . 5 ⊢ ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆))
12	8, 11	bitri 278	. . . 4 ⊢ (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆))
13	7, 12	bitr4di 292	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
14	3, 13	bitr2d 283	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
15		eqid 2758	. . . 4 ⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t 𝑆)
16	1, 15	restperf 21898	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
17	4, 16	sylan 583	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
18	1	clsss3 21773	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
19		eqid 2758	. . . . 5 ⊢ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) = (𝐽 ↾t ((cls‘𝐽)‘𝑆))
20	1, 19	restperf 21898	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
21	18, 20	syldan 594	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
22	4, 21	sylan 583	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
23	14, 17, 22	3bitr4d 314	1 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3858   ⊆ wss 3860  ∪ cuni 4801  ‘cfv 6340  (class class class)co 7156   ↾_t crest 16766  Topctop 21607  clsccl 21732  limPtclp 21848  Perfcperf 21849  Frect1 22021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-en 8541  df-fin 8544  df-fi 8921  df-rest 16768  df-topgen 16789  df-top 21608  df-topon 21625  df-bases 21660  df-cld 21733  df-ntr 21734  df-cls 21735  df-nei 21812  df-lp 21850  df-perf 21851  df-t1 22028

This theorem is referenced by: (None)