MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clsss2 Structured version   Visualization version   GIF version

Theorem clsss2 21675
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 21629 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
21adantr 484 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐽 ∈ Top)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43cldss 21632 . . . 4 (𝐶 ∈ (Clsd‘𝐽) → 𝐶𝑋)
54adantr 484 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝐶𝑋)
6 simpr 488 . . 3 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → 𝑆𝐶)
73clsss 21657 . . 3 ((𝐽 ∈ Top ∧ 𝐶𝑋𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
82, 5, 6, 7syl3anc 1368 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
9 cldcls 21645 . . 3 (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶)
109adantr 484 . 2 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶)
118, 10sseqtrd 3982 1 ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wss 3908   cuni 4813  cfv 6334  Topctop 21496  Clsdccld 21619  clsccl 21621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-top 21497  df-cld 21622  df-cls 21624
This theorem is referenced by:  elcls  21676  restcls  21784  cncls2i  21873  isnrm3  21962  lpcls  21967  isreg2  21980  dnsconst  21981  hauscmplem  22009  txcls  22207  ptclsg  22218  kqreglem1  22344  kqreglem2  22345  kqnrmlem1  22346  kqnrmlem2  22347  blcls  23111  clsocv  23852  resscdrg  23960  cldregopn  33753
  Copyright terms: Public domain W3C validator