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Theorem restcldr 23182
Description: A set which is closed in the subspace topology induced by a closed set is closed in the original topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
restcldr ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))

Proof of Theorem restcldr
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 cldrcl 23034 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 eqid 2737 . . . . 5 𝐽 = 𝐽
32cldss 23037 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
42restcld 23180 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
51, 3, 4syl2anc 584 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
6 incld 23051 . . . . . 6 ((𝑣 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑣𝐴) ∈ (Clsd‘𝐽))
76ancoms 458 . . . . 5 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝑣𝐴) ∈ (Clsd‘𝐽))
8 eleq1 2829 . . . . 5 (𝐵 = (𝑣𝐴) → (𝐵 ∈ (Clsd‘𝐽) ↔ (𝑣𝐴) ∈ (Clsd‘𝐽)))
97, 8syl5ibrcom 247 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝐵 = (𝑣𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
109rexlimdva 3155 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
115, 10sylbid 240 . 2 (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) → 𝐵 ∈ (Clsd‘𝐽)))
1211imp 406 1 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  cin 3950  wss 3951   cuni 4907  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027
This theorem is referenced by:  paste  23302  qtoprest  23725  zcld2  24837  sszcld  24839  logdmopn  26691  dvasin  37711  dvacos  37712  dvreasin  37713  dvreacos  37714
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