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Theorem clsss 20771
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 3591 . . . . . 6 (𝑇𝑆 → (𝑆𝑥𝑇𝑥))
21adantr 481 . . . . 5 ((𝑇𝑆𝑥 ∈ (Clsd‘𝐽)) → (𝑆𝑥𝑇𝑥))
32ss2rabdv 3664 . . . 4 (𝑇𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
4 intss 4465 . . . 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 1082 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
7 simp1 1059 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝐽 ∈ Top)
8 sstr2 3591 . . . . 5 (𝑇𝑆 → (𝑆𝑋𝑇𝑋))
98impcom 446 . . . 4 ((𝑆𝑋𝑇𝑆) → 𝑇𝑋)
1093adant1 1077 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑋)
11 clscld.1 . . . 4 𝑋 = 𝐽
1211clsval 20754 . . 3 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
137, 10, 12syl2anc 692 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
1411clsval 20754 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
15143adant3 1079 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
166, 13, 153sstr4d 3629 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  wss 3556   cuni 4404   cint 4442  cfv 5849  Topctop 20620  Clsdccld 20733  clsccl 20735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-top 20621  df-cld 20736  df-cls 20738
This theorem is referenced by:  ntrss  20772  clsss2  20789  lpsscls  20858  lpss3  20861  cnclsi  20989  cncls  20991  lpcls  21081  cnextcn  21784  clssubg  21825  clsnsg  21826  utopreg  21969  hauseqcn  29735  kur14lem6  30922  clsint2  31987  opnregcld  31988
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