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Theorem clsss2 20870
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 20824 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
21adantr 481 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐽 ∈ Top)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43cldss 20827 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐶𝑋)
54adantr 481 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐶𝑋)
6 simpr 477 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝑆𝐶)
73clsss 20852 . . 3 ((𝐽 ∈ Top ∧ 𝐶𝑋𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
82, 5, 6, 7syl3anc 1325 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
9 cldcls 20840 . . 3 (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶)
109adantr 481 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶)
118, 10sseqtrd 3639 1 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wss 3572   cuni 4434  cfv 5886  Topctop 20692  Clsdccld 20814  clsccl 20816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-top 20693  df-cld 20817  df-cls 20819
This theorem is referenced by:  elcls  20871  restcls  20979  cncls2i  21068  isnrm3  21157  lpcls  21162  isreg2  21175  dnsconst  21176  hauscmplem  21203  txcls  21401  ptclsg  21412  kqreglem1  21538  kqreglem2  21539  kqnrmlem1  21540  kqnrmlem2  21541  blcls  22305  clsocv  23043  resscdrg  23148  cldregopn  32310
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