MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  perfcls Structured version   Visualization version   GIF version
Theorem perfcls 22868
Description: A subset of a perfect space is perfect iff its closure is perfect (and the closure is an actual perfect set, since it is both closed and perfect in the subspace topology). (Contributed by Mario Carneiro, 26-Dec-2016.)
Hypothesis
Ref Expression
lpcls.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
perfcls ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝑆) ∈ Perf ↔ (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Perf))
Proof of Theorem perfcls
StepHypRef Expression
1 lpcls.1 . . . . 5 𝑋 = βˆͺ 𝐽
21lpcls 22867 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†))
32sseq2d 4014 . . 3 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) βŠ† ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† ((limPtβ€˜π½)β€˜π‘†)))
4 t1top 22833 . . . . . 6 (𝐽 ∈ Fre β†’ 𝐽 ∈ Top)
51clslp 22651 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
64, 5sylan 580 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
76sseq1d 4013 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) βŠ† ((limPtβ€˜π½)β€˜π‘†) ↔ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) βŠ† ((limPtβ€˜π½)β€˜π‘†)))
8 ssequn1 4180 . . . . 5 (𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†) ↔ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†))
9 ssun2 4173 . . . . . 6 ((limPtβ€˜π½)β€˜π‘†) βŠ† (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†))
10 eqss 3997 . . . . . 6 ((𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†) ↔ ((𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) βŠ† ((limPtβ€˜π½)β€˜π‘†) ∧ ((limPtβ€˜π½)β€˜π‘†) βŠ† (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†))))
119, 10mpbiran2 708 . . . . 5 ((𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†) ↔ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) βŠ† ((limPtβ€˜π½)β€˜π‘†))
128, 11bitri 274 . . . 4 (𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†) ↔ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) βŠ† ((limPtβ€˜π½)β€˜π‘†))
137, 12bitr4di 288 . . 3 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π‘†) βŠ† ((limPtβ€˜π½)β€˜π‘†) ↔ 𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†)))
143, 13bitr2d 279 . 2 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†) ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†))))
15 eqid 2732 . . . 4 (𝐽 β†Ύt 𝑆) = (𝐽 β†Ύt 𝑆)
161, 15restperf 22687 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝑆) ∈ Perf ↔ 𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†)))
174, 16sylan 580 . 2 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝑆) ∈ Perf ↔ 𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†)))
181clsss3 22562 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
19 eqid 2732 . . . . 5 (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) = (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†))
201, 19restperf 22687 . . . 4 ((𝐽 ∈ Top ∧ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋) β†’ ((𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Perf ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†))))
2118, 20syldan 591 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Perf ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†))))
224, 21sylan 580 . 2 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Perf ↔ ((clsβ€˜π½)β€˜π‘†) βŠ† ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†))))
2314, 17, 223bitr4d 310 1 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝑆) ∈ Perf ↔ (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) ∈ Perf))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3946   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7408   β†Ύt crest 17365  Topctop 22394  clsccl 22521  limPtclp 22637  Perfcperf 22638  Frect1 22810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-en 8939  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-cld 22522  df-ntr 22523  df-cls 22524  df-nei 22601  df-lp 22639  df-perf 22640  df-t1 22817
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator