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Theorem restlp 23074
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 = βˆͺ 𝐽
restcls.2 𝐾 = (𝐽 β†Ύt π‘Œ)
Assertion
Ref Expression
restlp ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ ((limPtβ€˜πΎ)β€˜π‘†) = (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ))
Proof of Theorem restlp
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simp3 1136 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† π‘Œ)
21ssdifssd 4138 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (𝑆 βˆ– {π‘₯}) βŠ† π‘Œ)
3 restcls.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
4 restcls.2 . . . . . . 7 𝐾 = (𝐽 β†Ύt π‘Œ)
53, 4restcls 23072 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ (𝑆 βˆ– {π‘₯}) βŠ† π‘Œ) β†’ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯})) = (((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∩ π‘Œ))
62, 5syld3an3 1407 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯})) = (((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∩ π‘Œ))
76eleq2d 2814 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯})) ↔ π‘₯ ∈ (((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∩ π‘Œ)))
8 elin 3960 . . . 4 (π‘₯ ∈ (((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∩ π‘Œ) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∧ π‘₯ ∈ π‘Œ))
97, 8bitrdi 287 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯})) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∧ π‘₯ ∈ π‘Œ)))
10 simp1 1134 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐽 ∈ Top)
113toptopon 22806 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
1210, 11sylib 217 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
13 simp2 1135 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ π‘Œ βŠ† 𝑋)
14 resttopon 23052 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
1512, 13, 14syl2anc 583 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
164, 15eqeltrid 2832 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
17 topontop 22802 . . . . 5 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
1816, 17syl 17 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐾 ∈ Top)
19 toponuni 22803 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
2016, 19syl 17 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
211, 20sseqtrd 4018 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† βˆͺ 𝐾)
22 eqid 2727 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
2322islp 23031 . . . 4 ((𝐾 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐾) β†’ (π‘₯ ∈ ((limPtβ€˜πΎ)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯}))))
2418, 21, 23syl2anc 583 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((limPtβ€˜πΎ)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜πΎ)β€˜(𝑆 βˆ– {π‘₯}))))
25 elin 3960 . . . 4 (π‘₯ ∈ (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ) ↔ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ∧ π‘₯ ∈ π‘Œ))
261, 13sstrd 3988 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† 𝑋)
273islp 23031 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
2810, 26, 27syl2anc 583 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
2928anbi1d 629 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ ((π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ∧ π‘₯ ∈ π‘Œ) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∧ π‘₯ ∈ π‘Œ)))
3025, 29bitrid 283 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯})) ∧ π‘₯ ∈ π‘Œ)))
319, 24, 303bitr4d 311 . 2 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (π‘₯ ∈ ((limPtβ€˜πΎ)β€˜π‘†) ↔ π‘₯ ∈ (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ)))
3231eqrdv 2725 1 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ ((limPtβ€˜πΎ)β€˜π‘†) = (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   βˆ– cdif 3941   ∩ cin 3943   βŠ† wss 3944  {csn 4624  βˆͺ cuni 4903  β€˜cfv 6542  (class class class)co 7414   β†Ύt crest 17393  Topctop 22782  TopOnctopon 22799  clsccl 22909  limPtclp 23025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-en 8956  df-fin 8959  df-fi 9426  df-rest 17395  df-topgen 17416  df-top 22783  df-topon 22800  df-bases 22836  df-cld 22910  df-cls 22912  df-lp 23027
This theorem is referenced by:  restperf  23075  lptioo2cn  44956  lptioo1cn  44957  limclner  44962  fourierdlem42  45460
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