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Theorem clsss2 23080
Description: If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clsss2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)

Proof of Theorem clsss2
StepHypRef Expression
1 cldrcl 23034 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
21adantr 480 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐽 ∈ Top)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43cldss 23037 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐶𝑋)
54adantr 480 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐶𝑋)
6 simpr 484 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝑆𝐶)
73clsss 23062 . . 3 ((𝐽 ∈ Top ∧ 𝐶𝑋𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
82, 5, 6, 7syl3anc 1373 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
9 cldcls 23050 . . 3 (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶)
109adantr 480 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶)
118, 10sseqtrd 4020 1 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wss 3951   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024  clsccl 23026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-cls 23029
This theorem is referenced by:  elcls  23081  restcls  23189  cncls2i  23278  isnrm3  23367  lpcls  23372  isreg2  23385  dnsconst  23386  hauscmplem  23414  txcls  23612  ptclsg  23623  kqreglem1  23749  kqreglem2  23750  kqnrmlem1  23751  kqnrmlem2  23752  blcls  24519  clsocv  25284  resscdrg  25392  cldregopn  36332  seposep  48823
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