Theorem lpval 20753

Description: The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
lpfval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
lpval	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem lpval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpfval.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	lpfval 20752	. . . 4 ⊢ (𝐽 ∈ Top → (limPt‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}))
3	2	fveq1d 6105	. . 3 ⊢ (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆))
4	3	adantr 480	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆))
5	1	topopn 20536	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6		elpw2g 4754	. . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
7	5, 6	syl 17	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
8	7	biimpar 501	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
9	5	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
10		ssdifss 3703	. . . . . 6 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
11	1	clsss3 20673	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ 𝑋)
12	11	sseld 3567	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ 𝑋))
13	10, 12	sylan2 490	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ 𝑋))
14	13	abssdv 3639	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ⊆ 𝑋)
15	9, 14	ssexd 4733	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ∈ V)
16		difeq1 3683	. . . . . . 7 ⊢ (𝑦 = 𝑆 → (𝑦 ∖ {𝑥}) = (𝑆 ∖ {𝑥}))
17	16	fveq2d 6107	. . . . . 6 ⊢ (𝑦 = 𝑆 → ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
18	17	eleq2d 2673	. . . . 5 ⊢ (𝑦 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
19	18	abbidv 2728	. . . 4 ⊢ (𝑦 = 𝑆 → {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))} = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
20		eqid 2610	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})
21	19, 20	fvmptg 6189	. . 3 ⊢ ((𝑆 ∈ 𝒫 𝑋 ∧ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ∈ V) → ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
22	8, 15, 21	syl2anc 691	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})
23	4, 22	eqtrd 2644	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ‘cfv 5804  Topctop 20517  clsccl 20632  limPtclp 20748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-cls 20635  df-lp 20750

This theorem is referenced by:  islp  20754  lpsscls  20755