Theorem perfcls 23303

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 23302	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
3	2	sseq2d 3991	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆)))
4		t1top 23268	. . . . . 6 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
5	1	clslp 23086	. . . . . 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 3990	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆)))
8		ssequn1 4161	. . . . 5 ⊢ (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
9		ssun2 4154	. . . . . 6 ⊢ ((limPt‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))
10		eqss 3974	. . . . . 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 2735	. . . 4 ⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t 𝑆)
16	1, 15	restperf 23122	. . 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 22997	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
19		eqid 2735	. . . . 5 ⊢ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) = (𝐽 ↾t ((cls‘𝐽)‘𝑆))
20	1, 19	restperf 23122	. . . 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 1540   ∈ wcel 2108   ∪ cun 3924   ⊆ wss 3926  ∪ cuni 4883  ‘cfv 6531  (class class class)co 7405   ↾_t crest 17434  Topctop 22831  clsccl 22956  limPtclp 23072  Perfcperf 23073  Frect1 23245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-en 8960  df-fin 8963  df-fi 9423  df-rest 17436  df-topgen 17457  df-top 22832  df-topon 22849  df-bases 22884  df-cld 22957  df-ntr 22958  df-cls 22959  df-nei 23036  df-lp 23074  df-perf 23075  df-t1 23252

This theorem is referenced by: (None)