Theorem clslp 23131

Description: The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)

Hypothesis

Ref	Expression
lpfval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

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

Proof of Theorem clslp

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpfval.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
2	1	neindisj 23100	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑥}))) → (𝑛 ∩ 𝑆) ≠ ∅)
3	2	expr 457	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ 𝑆) ≠ ∅))
4	3	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ 𝑆) ≠ ∅))
5		difsn 4731	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 𝑆 → (𝑆 ∖ {𝑥}) = 𝑆)
6	5	ineq2d 4149	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝑆 → (𝑛 ∩ (𝑆 ∖ {𝑥})) = (𝑛 ∩ 𝑆))
7	6	neeq1d 2993	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝑆 → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛 ∩ 𝑆) ≠ ∅))
8	7	adantl 482	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛 ∩ 𝑆) ≠ ∅))
9	4, 8	sylibrd 260	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
10	9	ex 413	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)))
11	10	ralrimdv 3137	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
12		simpll 772	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
13		simplr 774	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
14	1	clsss3 23042	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
15	14	sselda 3915	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ 𝑋)
16	1	islp2 23128	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
17	12, 13, 15, 16	syl3anc 1379	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
18	11, 17	sylibrd 260	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
19	18	orrd 869	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
20		elun 4083	. . . . 5 ⊢ (𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
21	19, 20	sylibr 235	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
22	21	ex 413	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
23	22	ssrdv 3921	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
24	1	sscls 23039	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
25	1	lpsscls 23124	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆))
26	24, 25	unssd 4121	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
27	23, 26	eqssd 3932	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  {csn 4555  ∪ cuni 4838  ‘cfv 6485  Topctop 22876  clsccl 23001  neicnei 23080  limPtclp 23117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-top 22877  df-cld 23002  df-ntr 23003  df-cls 23004  df-nei 23081  df-lp 23119

This theorem is referenced by:  islpi  23132  cldlp  23133  perfcls  23348