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Theorem restcldr 21198
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 21050 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 eqid 2770 . . . . 5 𝐽 = 𝐽
32cldss 21053 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
42restcld 21196 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
51, 3, 4syl2anc 565 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
6 incld 21067 . . . . . 6 ((𝑣 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑣𝐴) ∈ (Clsd‘𝐽))
76ancoms 455 . . . . 5 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝑣𝐴) ∈ (Clsd‘𝐽))
8 eleq1 2837 . . . . 5 (𝐵 = (𝑣𝐴) → (𝐵 ∈ (Clsd‘𝐽) ↔ (𝑣𝐴) ∈ (Clsd‘𝐽)))
97, 8syl5ibrcom 237 . . . 4 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝐵 = (𝑣𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
109rexlimdva 3178 . . 3 (𝐴 ∈ (Clsd‘𝐽) → (∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
115, 10sylbid 230 . 2 (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) → 𝐵 ∈ (Clsd‘𝐽)))
1211imp 393 1 ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wrex 3061  cin 3720  wss 3721   cuni 4572  cfv 6031  (class class class)co 6792  t crest 16288  Topctop 20917  Clsdccld 21040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-oadd 7716  df-er 7895  df-en 8109  df-fin 8112  df-fi 8472  df-rest 16290  df-topgen 16311  df-top 20918  df-topon 20935  df-bases 20970  df-cld 21043
This theorem is referenced by:  paste  21318  qtoprest  21740  zcld2  22837  sszcld  22839  logdmopn  24615  dvasin  33821  dvacos  33822  dvreasin  33823  dvreacos  33824
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