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Theorem restlp 23191
Description: The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
restcls.1 𝑋 = 𝐽
restcls.2 𝐾 = (𝐽t 𝑌)
Assertion
Ref Expression
restlp ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))

Proof of Theorem restlp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3 1139 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑌)
21ssdifssd 4147 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑆 ∖ {𝑥}) ⊆ 𝑌)
3 restcls.1 . . . . . . 7 𝑋 = 𝐽
4 restcls.2 . . . . . . 7 𝐾 = (𝐽t 𝑌)
53, 4restcls 23189 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
62, 5syld3an3 1411 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
76eleq2d 2827 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ 𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌)))
8 elin 3967 . . . 4 (𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌))
97, 8bitrdi 287 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
10 simp1 1137 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
113toptopon 22923 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
1210, 11sylib 218 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ (TopOn‘𝑋))
13 simp2 1138 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌𝑋)
14 resttopon 23169 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
1512, 13, 14syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
164, 15eqeltrid 2845 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ (TopOn‘𝑌))
17 topontop 22919 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
1816, 17syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
19 toponuni 22920 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2016, 19syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 = 𝐾)
211, 20sseqtrd 4020 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 𝐾)
22 eqid 2737 . . . . 5 𝐾 = 𝐾
2322islp 23148 . . . 4 ((𝐾 ∈ Top ∧ 𝑆 𝐾) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥}))))
2418, 21, 23syl2anc 584 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥}))))
25 elin 3967 . . . 4 (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥𝑌))
261, 13sstrd 3994 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
273islp 23148 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2810, 26, 27syl2anc 584 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
2928anbi1d 631 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
3025, 29bitrid 283 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
319, 24, 303bitr4d 311 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌)))
3231eqrdv 2735 1 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cdif 3948  cin 3950  wss 3951  {csn 4626   cuni 4907  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  TopOnctopon 22916  clsccl 23026  limPtclp 23142
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-cls 23029  df-lp 23144
This theorem is referenced by:  restperf  23192  lptioo2cn  45660  lptioo1cn  45661  limclner  45666  fourierdlem42  46164
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