Theorem restperf 22335

Description: Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.)

Hypotheses

Ref	Expression
restcls.1	⊢ 𝑋 = ∪ 𝐽
restcls.2	⊢ 𝐾 = (𝐽 ↾t 𝑌)

Assertion

Ref	Expression
restperf	⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))

Proof of Theorem restperf

Step	Hyp	Ref	Expression
1		restcls.2	. . . . 5 ⊢ 𝐾 = (𝐽 ↾t 𝑌)
2		restcls.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	toptopon 22066	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4		resttopon 22312	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
5	3, 4	sylanb 581	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
6	1, 5	eqeltrid 2843	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
7		topontop 22062	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
8	6, 7	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
9		eqid 2738	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
10	9	isperf 22302	. . . 4 ⊢ (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
11	10	baib 536	. . 3 ⊢ (𝐾 ∈ Top → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
12	8, 11	syl 17	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
13		sseqin2 4149	. . 3 ⊢ (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌)
14		ssid 3943	. . . . . 6 ⊢ 𝑌 ⊆ 𝑌
15	2, 1	restlp 22334	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
16	14, 15	mp3an3 1449	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
17		toponuni 22063	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
18	6, 17	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ 𝐾)
19	18	fveq2d 6778	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = ((limPt‘𝐾)‘∪ 𝐾))
20	16, 19	eqtr3d 2780	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = ((limPt‘𝐾)‘∪ 𝐾))
21	20, 18	eqeq12d 2754	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌 ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
22	13, 21	bitrid 282	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
23	12, 22	bitr4d 281	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887  ∪ cuni 4839  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131  Topctop 22042  TopOnctopon 22059  limPtclp 22285  Perfcperf 22286

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-cls 22172  df-lp 22287  df-perf 22288

This theorem is referenced by:  perfcls  22516  reperflem  23981  perfdvf  25067