Theorem clslp 23104

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 23073	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑥}))) → (𝑛 ∩ 𝑆) ≠ ∅)
3	2	expr 456	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ 𝑆) ≠ ∅))
4	3	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ 𝑆) ≠ ∅))
5		difsn 4756	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 𝑆 → (𝑆 ∖ {𝑥}) = 𝑆)
6	5	ineq2d 4174	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝑆 → (𝑛 ∩ (𝑆 ∖ {𝑥})) = (𝑛 ∩ 𝑆))
7	6	neeq1d 2992	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝑆 → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛 ∩ 𝑆) ≠ ∅))
8	7	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛 ∩ 𝑆) ≠ ∅))
9	4, 8	sylibrd 259	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
10	9	ex 412	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)))
11	10	ralrimdv 3136	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
12		simpll 767	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
13		simplr 769	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
14	1	clsss3 23015	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
15	14	sselda 3935	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ 𝑋)
16	1	islp2 23101	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
17	12, 13, 15, 16	syl3anc 1374	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))
18	11, 17	sylibrd 259	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
19	18	orrd 864	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
20		elun 4107	. . . . 5 ⊢ (𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ ((limPt‘𝐽)‘𝑆)))
21	19, 20	sylibr 234	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
22	21	ex 412	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
23	22	ssrdv 3941	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
24	1	sscls 23012	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
25	1	lpsscls 23097	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆))
26	24, 25	unssd 4146	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆))
27	23, 26	eqssd 3953	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582  ∪ cuni 4865  ‘cfv 6500  Topctop 22849  clsccl 22974  neicnei 23053  limPtclp 23090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22850  df-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054  df-lp 23092

This theorem is referenced by:  islpi  23105  cldlp  23106  perfcls  23321