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Theorem perfcls 23287
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 23286 . . . 4 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((limPtβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘†)) = ((limPtβ€˜π½)β€˜π‘†))
32sseq2d 4012 . . 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 23252 . . . . . 6 (𝐽 ∈ Fre β†’ 𝐽 ∈ Top)
51clslp 23070 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
64, 5sylan 578 . . . . 5 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
76sseq1d 4011 . . . 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 3995 . . . . . 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 2727 . . . 4 (𝐽 β†Ύt 𝑆) = (𝐽 β†Ύt 𝑆)
161, 15restperf 23106 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝑆) ∈ Perf ↔ 𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†)))
174, 16sylan 578 . 2 ((𝐽 ∈ Fre ∧ 𝑆 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝑆) ∈ Perf ↔ 𝑆 βŠ† ((limPtβ€˜π½)β€˜π‘†)))
181clsss3 22981 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
19 eqid 2727 . . . . 5 (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†)) = (𝐽 β†Ύt ((clsβ€˜π½)β€˜π‘†))
201, 19restperf 23106 . . . 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 589 . . 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 578 . 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 394   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3945   βŠ† wss 3947  βˆͺ cuni 4910  β€˜cfv 6551  (class class class)co 7424   β†Ύt crest 17407  Topctop 22813  clsccl 22940  limPtclp 23056  Perfcperf 23057  Frect1 23229
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-iin 5001  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-en 8969  df-fin 8972  df-fi 9440  df-rest 17409  df-topgen 17430  df-top 22814  df-topon 22831  df-bases 22867  df-cld 22941  df-ntr 22942  df-cls 22943  df-nei 23020  df-lp 23058  df-perf 23059  df-t1 23236
This theorem is referenced by: (None)
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