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Theorem restcldi 22897
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 1137 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘𝐽))
2 dfss 3966 . . . . 5 (𝐵𝐴𝐵 = (𝐵𝐴))
32biimpi 215 . . . 4 (𝐵𝐴𝐵 = (𝐵𝐴))
433ad2ant3 1135 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 = (𝐵𝐴))
5 ineq1 4205 . . . 4 (𝑣 = 𝐵 → (𝑣𝐴) = (𝐵𝐴))
65rspceeqv 3633 . . 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 22750 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
983ad2ant2 1134 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐽 ∈ Top)
10 simp1 1136 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐴𝑋)
11 restcldi.1 . . . 4 𝑋 = 𝐽
1211restcld 22896 . . 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 256 1 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087   = wceq 1541  wcel 2106  wrex 3070  cin 3947  wss 3948   cuni 4908  cfv 6543  (class class class)co 7411  t crest 17370  Topctop 22615  Clsdccld 22740
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17372  df-topgen 17393  df-top 22616  df-topon 22633  df-bases 22669  df-cld 22743
This theorem is referenced by:  txkgen  23376  qtoprest  23441  cnmpopc  24668  cnheiborlem  24694  abelth  26177  zarmxt1  33146  cvmliftlem10  34571  icccldii  47639
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