Theorem lpcls 23595

Description: The limit points of the closure of a subset are the same as the limit points of the set in a T₁ space. (Contributed by Mario Carneiro, 26-Dec-2016.)

Hypothesis

Ref	Expression
lpcls.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
lpcls	⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))

Proof of Theorem lpcls

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		t1top 23561	. . . . . . 7 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
2		lpcls.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
3	2	clsss3 23290	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
4	3	ssdifssd 4097	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
5	2	clsss3 23290	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
6	4, 5	syldan 603	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
7	1, 6	sylan 592	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
8	7	sseld 3933	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ 𝑋))
9		ssdifss 4090	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
10	2	clscld 23278	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
11	1, 9, 10	syl2an 608	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
12	11	adantr 486	. . . . . . . . 9 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
13	2	t1sncld 23557	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Fre ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (Clsd‘𝐽))
14	13	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (Clsd‘𝐽))
15		uncld 23272	. . . . . . . . . . . 12 ⊢ (({𝑥} ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽)) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
16	14, 12, 15	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
17	2	sscls 23287	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
18	1, 9, 17	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
19		ssundif 4446	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
20	18, 19	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
21	20	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
22	2	clsss2 23303	. . . . . . . . . . 11 ⊢ ((({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))) → ((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
23	16, 21, 22	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
24		ssundif 4446	. . . . . . . . . 10 ⊢ (((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
25	23, 24	sylib 221	. . . . . . . . 9 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
26	2	clsss2 23303	. . . . . . . . 9 ⊢ ((((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽) ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
27	12, 25, 26	syl2anc 596	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
28	27	sseld 3933	. . . . . . 7 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
29	28	ex 418	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝑋 → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))))
30	29	com23 87	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → (𝑥 ∈ 𝑋 → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))))
31	8, 30	mpdd 44	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
32	1	adantr 486	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
33	1, 3	sylan 592	. . . . . . 7 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
34	33	ssdifssd 4097	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
35	2	sscls 23287	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
36	1, 35	sylan 592	. . . . . . 7 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
37	36	ssdifd 4095	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ (((cls‘𝐽)‘𝑆) ∖ {𝑥}))
38	2	clsss 23285	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ (((cls‘𝐽)‘𝑆) ∖ {𝑥})) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})))
39	32, 34, 37, 38	syl3anc 1398	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})))
40	39	sseld 3933	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
41	31, 40	impbid 215	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
42	2	islp 23371	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
43	3, 42	syldan 603	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
44	1, 43	sylan 592	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
45	2	islp 23371	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
46	1, 45	sylan 592	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
47	41, 44, 46	3bitr4d 314	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
48	47	eqrdv 2760	1 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  {csn 4587  ∪ cuni 4870  ‘cfv 6537  Topctop 23124  Clsdccld 23247  clsccl 23249  limPtclp 23365  Frect1 23538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23125  df-cld 23250  df-cls 23252  df-lp 23367  df-t1 23545

This theorem is used by:  perfcls  23596