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Theorem restcldi 23196
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 1136 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘𝐽))
2 dfss 3981 . . . . 5 (𝐵𝐴𝐵 = (𝐵𝐴))
32biimpi 216 . . . 4 (𝐵𝐴𝐵 = (𝐵𝐴))
433ad2ant3 1134 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 = (𝐵𝐴))
5 ineq1 4220 . . . 4 (𝑣 = 𝐵 → (𝑣𝐴) = (𝐵𝐴))
65rspceeqv 3644 . . 3 ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 = (𝐵𝐴)) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴))
71, 4, 6syl2anc 584 . 2 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴))
8 cldrcl 23049 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
983ad2ant2 1133 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐽 ∈ Top)
10 simp1 1135 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐴𝑋)
11 restcldi.1 . . . 4 𝑋 = 𝐽
1211restcld 23195 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
139, 10, 12syl2anc 584 . 2 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
147, 13mpbird 257 1 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1536  wcel 2105  wrex 3067  cin 3961  wss 3962   cuni 4911  cfv 6562  (class class class)co 7430  t crest 17466  Topctop 22914  Clsdccld 23039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-en 8984  df-fin 8987  df-fi 9448  df-rest 17468  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-cld 23042
This theorem is referenced by:  txkgen  23675  qtoprest  23740  cnmpopc  24968  cnheiborlem  24999  abelth  26499  zarmxt1  33840  cvmliftlem10  35278  icccldii  48714
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