MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restperf Structured version   Visualization version   GIF version
Theorem restperf 23043
Description: Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (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
restperf ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (𝐾 ∈ Perf ↔ π‘Œ βŠ† ((limPtβ€˜π½)β€˜π‘Œ)))
Proof of Theorem restperf
StepHypRef Expression
1 restcls.2 . . . . 5 𝐾 = (𝐽 β†Ύt π‘Œ)
2 restcls.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
32toptopon 22774 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
4 resttopon 23020 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
53, 4sylanb 580 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) ∈ (TopOnβ€˜π‘Œ))
61, 5eqeltrid 2831 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
7 topontop 22770 . . . 4 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
86, 7syl 17 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ 𝐾 ∈ Top)
9 eqid 2726 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
109isperf 23010 . . . 4 (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
1110baib 535 . . 3 (𝐾 ∈ Top β†’ (𝐾 ∈ Perf ↔ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
128, 11syl 17 . 2 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (𝐾 ∈ Perf ↔ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
13 sseqin2 4210 . . 3 (π‘Œ βŠ† ((limPtβ€˜π½)β€˜π‘Œ) ↔ (((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ) = π‘Œ)
14 ssid 3999 . . . . . 6 π‘Œ βŠ† π‘Œ
152, 1restlp 23042 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋 ∧ π‘Œ βŠ† π‘Œ) β†’ ((limPtβ€˜πΎ)β€˜π‘Œ) = (((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ))
1614, 15mp3an3 1446 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ ((limPtβ€˜πΎ)β€˜π‘Œ) = (((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ))
17 toponuni 22771 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
186, 17syl 17 . . . . . 6 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ π‘Œ = βˆͺ 𝐾)
1918fveq2d 6889 . . . . 5 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ ((limPtβ€˜πΎ)β€˜π‘Œ) = ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾))
2016, 19eqtr3d 2768 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ) = ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾))
2120, 18eqeq12d 2742 . . 3 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ ((((limPtβ€˜π½)β€˜π‘Œ) ∩ π‘Œ) = π‘Œ ↔ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
2213, 21bitrid 283 . 2 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (π‘Œ βŠ† ((limPtβ€˜π½)β€˜π‘Œ) ↔ ((limPtβ€˜πΎ)β€˜βˆͺ 𝐾) = βˆͺ 𝐾))
2312, 22bitr4d 282 1 ((𝐽 ∈ Top ∧ π‘Œ βŠ† 𝑋) β†’ (𝐾 ∈ Perf ↔ π‘Œ βŠ† ((limPtβ€˜π½)β€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ∩ cin 3942   βŠ† wss 3943  βˆͺ cuni 4902  β€˜cfv 6537  (class class class)co 7405   β†Ύt crest 17375  Topctop 22750  TopOnctopon 22767  limPtclp 22993  Perfcperf 22994
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 7722
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 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17377  df-topgen 17398  df-top 22751  df-topon 22768  df-bases 22804  df-cld 22878  df-cls 22880  df-lp 22995  df-perf 22996
This theorem is referenced by:  perfcls  23224  reperflem  24689  perfdvf  25787
  Copyright terms: Public domain W3C validator