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Theorem restcldi 21465
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 1130 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘𝐽))
2 dfss 3875 . . . . 5 (𝐵𝐴𝐵 = (𝐵𝐴))
32biimpi 217 . . . 4 (𝐵𝐴𝐵 = (𝐵𝐴))
433ad2ant3 1128 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 = (𝐵𝐴))
5 ineq1 4101 . . . 4 (𝑣 = 𝐵 → (𝑣𝐴) = (𝐵𝐴))
65rspceeqv 3577 . . 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 21318 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
983ad2ant2 1127 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐽 ∈ Top)
10 simp1 1129 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐴𝑋)
11 restcldi.1 . . . 4 𝑋 = 𝐽
1211restcld 21464 . . 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 258 1 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1080   = wceq 1522  wcel 2081  wrex 3106  cin 3858  wss 3859   cuni 4745  cfv 6225  (class class class)co 7016  t crest 16523  Topctop 21185  Clsdccld 21308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-oadd 7957  df-er 8139  df-en 8358  df-fin 8361  df-fi 8721  df-rest 16525  df-topgen 16546  df-top 21186  df-topon 21203  df-bases 21238  df-cld 21311
This theorem is referenced by:  txkgen  21944  qtoprest  22009  cnmpopc  23215  cnheiborlem  23241  abelth  24712  cvmliftlem10  32149
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