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Theorem restlp 22687
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 1139 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† π‘Œ)
21ssdifssd 4143 . . . . . 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 22685 . . . . . 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 1410 . . . . 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 2820 . . . 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 3965 . . . 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 1137 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐽 ∈ Top)
113toptopon 22419 . . . . . . . 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 1138 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ π‘Œ βŠ† 𝑋)
14 resttopon 22665 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
1512, 13, 14syl2anc 585 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
164, 15eqeltrid 2838 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
17 topontop 22415 . . . . 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 22416 . . . . . 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 4023 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† βˆͺ 𝐾)
22 eqid 2733 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
2322islp 22644 . . . 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 585 . . 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 3965 . . . 4 (π‘₯ ∈ (((limPtβ€˜π½)β€˜π‘†) ∩ π‘Œ) ↔ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ∧ π‘₯ ∈ π‘Œ))
261, 13sstrd 3993 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ 𝑆 βŠ† π‘Œ) β†’ 𝑆 βŠ† 𝑋)
273islp 22644 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π‘†) ↔ π‘₯ ∈ ((clsβ€˜π½)β€˜(𝑆 βˆ– {π‘₯}))))
2810, 26, 27syl2anc 585 . . . . 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 631 . . . 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 2731 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  Topctop 22395  TopOnctopon 22412  clsccl 22522  limPtclp 22638
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-en 8940  df-fin 8943  df-fi 9406  df-rest 17368  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-cls 22525  df-lp 22640
This theorem is referenced by:  restperf  22688  lptioo2cn  44409  lptioo1cn  44410  limclner  44415  fourierdlem42  44913
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