Theorem restperf 23410

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 23143	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4		resttopon 23387	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
5	3, 4	sylanb 593	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
6	1, 5	eqeltrid 2866	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
7		topontop 23139	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
8	6, 7	syl 18	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
9		eqid 2762	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
10	9	isperf 23377	. . . 4 ⊢ (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
11	10	baib 545	. . 3 ⊢ (𝐾 ∈ Top → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
12	8, 11	syl 18	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
13		sseqin2 4172	. . 3 ⊢ (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌)
14		ssid 3956	. . . . . 6 ⊢ 𝑌 ⊆ 𝑌
15	2, 1	restlp 23409	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
16	14, 15	mp3an3 1479	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌))
17		toponuni 23140	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
18	6, 17	syl 18	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ 𝐾)
19	18	fveq2d 6886	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = ((limPt‘𝐾)‘∪ 𝐾))
20	16, 19	eqtr3d 2799	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = ((limPt‘𝐾)‘∪ 𝐾))
21	20, 18	eqeq12d 2778	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌 ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
22	13, 21	bitrid 286	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾))
23	12, 22	bitr4d 285	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4870  ‘cfv 6537  (class class class)co 7416   ↾_t crest 17509  Topctop 23119  TopOnctopon 23136  limPtclp 23360  Perfcperf 23361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-en 8956  df-fin 8959  df-fi 9384  df-rest 17511  df-topgen 17532  df-top 23120  df-topon 23137  df-bases 23172  df-cld 23245  df-cls 23247  df-lp 23362  df-perf 23363

This theorem is used by:  perfcls  23591  reperflem  25046  perfdvf  26132