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Theorem restlp 21719
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 1130 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑌)
21ssdifssd 4116 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑆 ∖ {𝑥}) ⊆ 𝑌)
3 restcls.1 . . . . . . 7 𝑋 = 𝐽
4 restcls.2 . . . . . . 7 𝐾 = (𝐽t 𝑌)
53, 4restcls 21717 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
62, 5syld3an3 1401 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))
76eleq2d 2895 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ 𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌)))
8 elin 4166 . . . 4 (𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌))
97, 8syl6bb 288 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
10 simp1 1128 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
113toptopon 21453 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
1210, 11sylib 219 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ (TopOn‘𝑋))
13 simp2 1129 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌𝑋)
14 resttopon 21697 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
1512, 13, 14syl2anc 584 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ (TopOn‘𝑌))
164, 15eqeltrid 2914 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ (TopOn‘𝑌))
17 topontop 21449 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
1816, 17syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
19 toponuni 21450 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2016, 19syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 = 𝐾)
211, 20sseqtrd 4004 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 𝐾)
22 eqid 2818 . . . . 5 𝐾 = 𝐾
2322islp 21676 . . . 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 4166 . . . 4 (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥𝑌))
261, 13sstrd 3974 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
273islp 21676 . . . . . 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 629 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
3025, 29syl5bb 284 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥𝑌)))
319, 24, 303bitr4d 312 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌)))
3231eqrdv 2816 1 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  cdif 3930  cin 3932  wss 3933  {csn 4557   cuni 4830  cfv 6348  (class class class)co 7145  t crest 16682  Topctop 21429  TopOnctopon 21446  clsccl 21554  limPtclp 21670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-er 8278  df-en 8498  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-cls 21557  df-lp 21672
This theorem is referenced by:  restperf  21720  lptioo2cn  41802  lptioo1cn  41803  limclner  41808  fourierdlem42  42311
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