Theorem perfcls 23273

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 23272	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
3	2	sseq2d 3965	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆)))
4		t1top 23238	. . . . . 6 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
5	1	clslp 23056	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
6	4, 5	sylan 580	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
7	6	sseq1d 3964	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆)))
8		ssequn1 4134	. . . . 5 ⊢ (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
9		ssun2 4127	. . . . . 6 ⊢ ((limPt‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))
10		eqss 3948	. . . . . 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 710	. . . . 5 ⊢ ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆))
12	8, 11	bitri 275	. . . 4 ⊢ (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆))
13	7, 12	bitr4di 289	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
14	3, 13	bitr2d 280	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
15		eqid 2730	. . . 4 ⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t 𝑆)
16	1, 15	restperf 23092	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
17	4, 16	sylan 580	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆)))
18	1	clsss3 22967	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
19		eqid 2730	. . . . 5 ⊢ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) = (𝐽 ↾t ((cls‘𝐽)‘𝑆))
20	1, 19	restperf 23092	. . . 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 591	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
22	4, 21	sylan 580	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆))))
23	14, 17, 22	3bitr4d 311	1 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ∪ cun 3898   ⊆ wss 3900  ∪ cuni 4857  ‘cfv 6477  (class class class)co 7341   ↾_t crest 17316  Topctop 22801  clsccl 22926  limPtclp 23042  Perfcperf 23043  Frect1 23215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-en 8865  df-fin 8868  df-fi 9290  df-rest 17318  df-topgen 17339  df-top 22802  df-topon 22819  df-bases 22854  df-cld 22927  df-ntr 22928  df-cls 22929  df-nei 23006  df-lp 23044  df-perf 23045  df-t1 23222

This theorem is referenced by: (None)