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Theorem clsss 23078
Description: Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clsss ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem clsss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sstr2 4002 . . . . . 6 (𝑇𝑆 → (𝑆𝑥𝑇𝑥))
21adantr 480 . . . . 5 ((𝑇𝑆𝑥 ∈ (Clsd‘𝐽)) → (𝑆𝑥𝑇𝑥))
32ss2rabdv 4086 . . . 4 (𝑇𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
4 intss 4974 . . . 4 ({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
53, 4syl 17 . . 3 (𝑇𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
653ad2ant3 1134 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
7 simp1 1135 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝐽 ∈ Top)
8 sstr2 4002 . . . . 5 (𝑇𝑆 → (𝑆𝑋𝑇𝑋))
98impcom 407 . . . 4 ((𝑆𝑋𝑇𝑆) → 𝑇𝑋)
1093adant1 1129 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑋)
11 clscld.1 . . . 4 𝑋 = 𝐽
1211clsval 23061 . . 3 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
137, 10, 12syl2anc 584 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
1411clsval 23061 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
15143adant3 1131 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
166, 13, 153sstr4d 4043 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  wss 3963   cuni 4912   cint 4951  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
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:  ntrss  23079  clsss2  23096  lpsscls  23165  lpss3  23168  cnclsi  23296  cncls  23298  lpcls  23388  cnextcn  24091  clssubg  24133  clsnsg  24134  utopreg  24277  hauseqcn  33859  kur14lem6  35196  clsint2  36312  opnregcld  36313
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