Theorem clsss2 23037

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) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)

Proof of Theorem clsss2

Step	Hyp	Ref	Expression
1		cldrcl 22991	. . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2	1	adantr 479	. . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐽 ∈ Top)
3		clscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	3	cldss 22994	. . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ 𝑋)
5	4	adantr 479	. . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐶 ⊆ 𝑋)
6		simpr 483	. . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝑆 ⊆ 𝐶)
7	3	clsss 23019	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
8	2, 5, 6, 7	syl3anc 1368	. 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶))
9		cldcls 23007	. . 3 ⊢ (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶)
10	9	adantr 479	. 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶)
11	8, 10	sseqtrd 4017	1 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ⊆ wss 3944  ∪ cuni 4909  ‘cfv 6549  Topctop 22856  Clsdccld 22981  clsccl 22983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-top 22857  df-cld 22984  df-cls 22986

This theorem is referenced by:  elcls  23038  restcls  23146  cncls2i  23235  isnrm3  23324  lpcls  23329  isreg2  23342  dnsconst  23343  hauscmplem  23371  txcls  23569  ptclsg  23580  kqreglem1  23706  kqreglem2  23707  kqnrmlem1  23708  kqnrmlem2  23709  blcls  24476  clsocv  25239  resscdrg  25347  cldregopn  35966  seposep  48135