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Theorem perfcls 23220
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 23219 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†))
32sseq2d 4009 . . 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 23185 . . . . . 6 (𝐽 ∈ Fre β†’ 𝐽 ∈ Top)
51clslp 23003 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
64, 5sylan 579 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
76sseq1d 4008 . . . 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 4175 . . . . 5 (𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†) ↔ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†))
9 ssun2 4168 . . . . . 6 ((limPtβ€˜π½)β€˜π‘†) βŠ† (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†))
10 eqss 3992 . . . . . 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 707 . . . . 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 275 . . . 4 (𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†) ↔ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) βŠ† ((limPtβ€˜π½)β€˜π‘†))
137, 12bitr4di 289 . . 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 280 . 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 2726 . . . 4 (𝐽 β†Ύt 𝑆) = (𝐽 β†Ύt 𝑆)
161, 15restperf 23039 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝑆) ∈ Perf ↔ 𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†)))
174, 16sylan 579 . 2 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝑆) ∈ Perf ↔ 𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†)))
181clsss3 22914 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
19 eqid 2726 . . . . 5 (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) = (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†))
201, 19restperf 23039 . . . 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 590 . . 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 579 . 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 311 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 395   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3941   βŠ† wss 3943  βˆͺ cuni 4902  β€˜cfv 6536  (class class class)co 7404   β†Ύt crest 17373  Topctop 22746  clsccl 22873  limPtclp 22989  Perfcperf 22990  Frect1 23162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-en 8939  df-fin 8942  df-fi 9405  df-rest 17375  df-topgen 17396  df-top 22747  df-topon 22764  df-bases 22800  df-cld 22874  df-ntr 22875  df-cls 22876  df-nei 22953  df-lp 22991  df-perf 22992  df-t1 23169
This theorem is referenced by: (None)
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