Theorem lpsscls 22038

Description: The limit points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007.)

Hypothesis

Ref	Expression
lpfval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
lpsscls	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem lpsscls

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpfval.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	lpval 22036	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
3		difss 4046	. . . . 5 ⊢ (𝑆 ∖ {𝑥}) ⊆ 𝑆
4	1	clsss 21951	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑆) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘𝑆))
5	3, 4	mp3an3 1452	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘𝑆))
6	5	sseld 3900	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
7	6	abssdv 3982	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ⊆ ((cls‘𝐽)‘𝑆))
8	2, 7	eqsstrd 3939	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  {cab 2714   ∖ cdif 3863   ⊆ wss 3866  {csn 4541  ∪ cuni 4819  ‘cfv 6380  Topctop 21790  clsccl 21915  limPtclp 22031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-top 21791  df-cld 21916  df-cls 21918  df-lp 22033

This theorem is referenced by:  lpss  22039  clslp  22045