Theorem lpcls 21168

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 21134	. . . . . . 7 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
2		lpcls.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
3	2	clsss3 20863	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
4	3	ssdifssd 3748	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
5	2	clsss3 20863	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
6	4, 5	syldan 487	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
7	1, 6	sylan 488	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ 𝑋)
8	7	sseld 3602	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ 𝑋))
9		ssdifss 3741	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
10	2	clscld 20851	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
11	1, 9, 10	syl2an 494	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
12	11	adantr 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽))
13	2	t1sncld 21130	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Fre ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (Clsd‘𝐽))
14	13	adantlr 751	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (Clsd‘𝐽))
15		uncld 20845	. . . . . . . . . . . 12 ⊢ (({𝑥} ∈ (Clsd‘𝐽) ∧ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∈ (Clsd‘𝐽)) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
16	14, 12, 15	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ∈ (Clsd‘𝐽))
17	2	sscls 20860	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
18	1, 9, 17	syl2an 494	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
19		ssundif 4052	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
20	18, 19	sylibr 224	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
21	20	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑆 ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
22	2	clsss2 20876	. . . . . . . . . . 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 693	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
24		ssundif 4052	. . . . . . . . . 10 ⊢ (((cls‘𝐽)‘𝑆) ⊆ ({𝑥} ∪ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) ↔ (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
25	23, 24	sylib 208	. . . . . . . . 9 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
26	2	clsss2 20876	. . . . . . . . 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 693	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))
28	27	sseld 3602	. . . . . . 7 ⊢ (((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
29	28	ex 450	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ 𝑋 → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))))
30	29	com23 86	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → (𝑥 ∈ 𝑋 → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))))
31	8, 30	mpdd 43	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
32	1	adantr 481	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
33	1, 3	sylan 488	. . . . . . 7 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
34	33	ssdifssd 3748	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ∖ {𝑥}) ⊆ 𝑋)
35	2	sscls 20860	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
36	1, 35	sylan 488	. . . . . . 7 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
37	36	ssdifd 3746	. . . . . 6 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ {𝑥}) ⊆ (((cls‘𝐽)‘𝑆) ∖ {𝑥}))
38	2	clsss 20858	. . . . . 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 1326	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})))
40	39	sseld 3602	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
41	31, 40	impbid 202	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
42	2	islp 20944	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
43	3, 42	syldan 487	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
44	1, 43	sylan 488	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((cls‘𝐽)‘(((cls‘𝐽)‘𝑆) ∖ {𝑥}))))
45	2	islp 20944	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
46	1, 45	sylan 488	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))))
47	41, 44, 46	3bitr4d 300	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
48	47	eqrdv 2620	1 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  {csn 4177  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  Clsdccld 20820  clsccl 20822  limPtclp 20938  Frect1 21111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-cls 20825  df-lp 20940  df-t1 21118

This theorem is referenced by:  perfcls  21169