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Theorem clsss2 21078
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 21032 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
21adantr 472 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐽 ∈ Top)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43cldss 21035 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐶𝑋)
54adantr 472 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐶𝑋)
6 simpr 479 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝑆𝐶)
73clsss 21060 . . 3 ((𝐽 ∈ Top ∧ 𝐶𝑋𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
82, 5, 6, 7syl3anc 1477 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
9 cldcls 21048 . . 3 (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶)
109adantr 472 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶)
118, 10sseqtrd 3782 1 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wss 3715   cuni 4588  cfv 6049  Topctop 20900  Clsdccld 21022  clsccl 21024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-cls 21027
This theorem is referenced by:  elcls  21079  restcls  21187  cncls2i  21276  isnrm3  21365  lpcls  21370  isreg2  21383  dnsconst  21384  hauscmplem  21411  txcls  21609  ptclsg  21620  kqreglem1  21746  kqreglem2  21747  kqnrmlem1  21748  kqnrmlem2  21749  blcls  22512  clsocv  23249  resscdrg  23354  cldregopn  32632
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