Theorem clsss 22992

Description: Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsss	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem clsss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3965	. . . . . 6 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥))
2	1	adantr 480	. . . . 5 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥))
3	2	ss2rabdv 4051	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
4		intss 4945	. . . 4 ⊢ ({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
5	3, 4	syl 17	. . 3 ⊢ (𝑇 ⊆ 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
6	5	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
7		simp1 1136	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top)
8		sstr2 3965	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑋 → 𝑇 ⊆ 𝑋))
9	8	impcom 407	. . . 4 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
10	9	3adant1 1130	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
11		clscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
12	11	clsval 22975	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
13	7, 10, 12	syl2anc 584	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
14	11	clsval 22975	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
15	14	3adant3 1132	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
16	6, 13, 15	3sstr4d 4014	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  {crab 3415   ⊆ wss 3926  ∪ cuni 4883  ∩ cint 4922  ‘cfv 6531  Topctop 22831  Clsdccld 22954  clsccl 22956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-top 22832  df-cld 22957  df-cls 22959

This theorem is referenced by:  ntrss  22993  clsss2  23010  lpsscls  23079  lpss3  23082  cnclsi  23210  cncls  23212  lpcls  23302  cnextcn  24005  clssubg  24047  clsnsg  24048  utopreg  24191  hauseqcn  33929  kur14lem6  35233  clsint2  36347  opnregcld  36348