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Theorem restcldr 21887
 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 21739 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 eqid 2758 . . . . 5 𝐽 = 𝐽
32cldss 21742 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
42restcld 21885 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
51, 3, 4syl2anc 587 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
6 incld 21756 . . . . . 6 ((𝑣 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑣𝐴) ∈ (Clsd‘𝐽))
76ancoms 462 . . . . 5 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝑣𝐴) ∈ (Clsd‘𝐽))
8 eleq1 2839 . . . . 5 (𝐵 = (𝑣𝐴) → (𝐵 ∈ (Clsd‘𝐽) ↔ (𝑣𝐴) ∈ (Clsd‘𝐽)))
97, 8syl5ibrcom 250 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝐵 = (𝑣𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
109rexlimdva 3208 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
115, 10sylbid 243 . 2 (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) → 𝐵 ∈ (Clsd‘𝐽)))
1211imp 410 1 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ∩ cin 3859   ⊆ wss 3860  ∪ cuni 4801  ‘cfv 6340  (class class class)co 7156   ↾t crest 16765  Topctop 21606  Clsdccld 21729 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-en 8541  df-fin 8544  df-fi 8921  df-rest 16767  df-topgen 16788  df-top 21607  df-topon 21624  df-bases 21659  df-cld 21732 This theorem is referenced by:  paste  22007  qtoprest  22430  zcld2  23529  sszcld  23531  logdmopn  25352  dvasin  35455  dvacos  35456  dvreasin  35457  dvreacos  35458
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