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Theorem clsss2 23096
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 23050 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
21adantr 480 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐽 ∈ Top)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43cldss 23053 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐶𝑋)
54adantr 480 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐶𝑋)
6 simpr 484 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝑆𝐶)
73clsss 23078 . . 3 ((𝐽 ∈ Top ∧ 𝐶𝑋𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
82, 5, 6, 7syl3anc 1370 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
9 cldcls 23066 . . 3 (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶)
109adantr 480 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶)
118, 10sseqtrd 4036 1 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wss 3963   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  clsccl 23042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-cls 23045
This theorem is referenced by:  elcls  23097  restcls  23205  cncls2i  23294  isnrm3  23383  lpcls  23388  isreg2  23401  dnsconst  23402  hauscmplem  23430  txcls  23628  ptclsg  23639  kqreglem1  23765  kqreglem2  23766  kqnrmlem1  23767  kqnrmlem2  23768  blcls  24535  clsocv  25298  resscdrg  25406  cldregopn  36314  seposep  48722
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