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Theorem restcldi 21886
 Description: A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
restcldi.1 𝑋 = 𝐽
Assertion
Ref Expression
restcldi ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))

Proof of Theorem restcldi
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 simp2 1134 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘𝐽))
2 dfss 3878 . . . . 5 (𝐵𝐴𝐵 = (𝐵𝐴))
32biimpi 219 . . . 4 (𝐵𝐴𝐵 = (𝐵𝐴))
433ad2ant3 1132 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 = (𝐵𝐴))
5 ineq1 4111 . . . 4 (𝑣 = 𝐵 → (𝑣𝐴) = (𝐵𝐴))
65rspceeqv 3558 . . 3 ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 = (𝐵𝐴)) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴))
71, 4, 6syl2anc 587 . 2 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴))
8 cldrcl 21739 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
983ad2ant2 1131 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐽 ∈ Top)
10 simp1 1133 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐴𝑋)
11 restcldi.1 . . . 4 𝑋 = 𝐽
1211restcld 21885 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
139, 10, 12syl2anc 587 . 2 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
147, 13mpbird 260 1 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = 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-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:  txkgen  22365  qtoprest  22430  cnmpopc  23642  cnheiborlem  23668  abelth  25148  zarmxt1  31363  cvmliftlem10  32784  icccldii  45651
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