Theorem lpval 9019

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.

Hypothesis

Ref	Expression
lpfval.1	|- X = U.J

Assertion

Ref	Expression
lpval	|- ((J e. Top /\ S C_ X) -> ((limPt` J)` S) = {x | x e. ((cls` J)` (S \ {x}))})

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

Proof of Theorem lpval

Step	Hyp	Ref	Expression
1		lpfval.1	. . . . . 6 |- X = U.J
2	1	lpfval 9018	. . . . 5 |- (J e. Top -> (limPt` J) = {<.y, z>. | (y C_ X /\ z = {x | x e. ((cls` J)` (y \ {x}))})})
3	2	adantr 425	. . . 4 |- ((J e. Top /\ S C_ X) -> (limPt` J) = {<.y, z>. | (y C_ X /\ z = {x | x e. ((cls` J)` (y \ {x}))})})
4		visset 2295	. . . . . . 7 |- y e. _V
5	4	elpw 3037	. . . . . 6 |- (y e. ~PX <-> y C_ X)
6	5	anbi1i 539	. . . . 5 |- ((y e. ~PX /\ z = {x | x e. ((cls` J)` (y \ {x}))}) <-> (y C_ X /\ z = {x | x e. ((cls` J)` (y \ {x}))}))
7	6	opabbii 3402	. . . 4 |- {<.y, z>. | (y e. ~PX /\ z = {x | x e. ((cls` J)` (y \ {x}))})} = {<.y, z>. | (y C_ X /\ z = {x | x e. ((cls` J)` (y \ {x}))})}
8	3, 7	syl6eqr 1946	. . 3 |- ((J e. Top /\ S C_ X) -> (limPt` J) = {<.y, z>. | (y e. ~PX /\ z = {x | x e. ((cls` J)` (y \ {x}))})})
9	8	fveq1d 4683	. 2 |- ((J e. Top /\ S C_ X) -> ((limPt` J)` S) = ({<.y, z>. | (y e. ~PX /\ z = {x | x e. ((cls` J)` (y \ {x}))})}` S))
10		elpw2g 3463	. . . . 5 |- (X e. _V -> (S e. ~PX <-> S C_ X))
11	10	biimpar 461	. . . 4 |- ((X e. _V /\ S C_ X) -> S e. ~PX)
12		uniexg 3795	. . . . 5 |- (J e. Top -> U.J e. _V)
13	12, 1	syl5eqel 1975	. . . 4 |- (J e. Top -> X e. _V)
14	11, 13	sylan 497	. . 3 |- ((J e. Top /\ S C_ X) -> S e. ~PX)
15		difss 2735	. . . . . . 7 |- (S \ {x}) C_ S
16	1	clsss 8963	. . . . . . 7 |- ((J e. Top /\ S C_ X /\ (S \ {x}) C_ S) -> ((cls` J)` (S \ {x})) C_ ((cls` J)` S))
17	15, 16	mp3an3 1180	. . . . . 6 |- ((J e. Top /\ S C_ X) -> ((cls` J)` (S \ {x})) C_ ((cls` J)` S))
18	17	sseld 2619	. . . . 5 |- ((J e. Top /\ S C_ X) -> (x e. ((cls` J)` (S \ {x})) -> x e. ((cls` J)` S)))
19	18	ss2abdv 2680	. . . 4 |- ((J e. Top /\ S C_ X) -> {x | x e. ((cls` J)` (S \ {x}))} C_ {x | x e. ((cls` J)` S)})
20		abid2 2011	. . . . . 6 |- {x | x e. ((cls` J)` S)} = ((cls` J)` S)
21		fvex 4689	. . . . . 6 |- ((cls` J)` S) e. _V
22	20, 21	eqeltri 1967	. . . . 5 |- {x | x e. ((cls` J)` S)} e. _V
23	22	ssex 3455	. . . 4 |- ({x | x e. ((cls` J)` (S \ {x}))} C_ {x | x e. ((cls` J)` S)} -> {x | x e. ((cls` J)` (S \ {x}))} e. _V)
24	19, 23	syl 12	. . 3 |- ((J e. Top /\ S C_ X) -> {x | x e. ((cls` J)` (S \ {x}))} e. _V)
25		difeq1 2717	. . . . . . 7 |- (y = S -> (y \ {x}) = (S \ {x}))
26	25	fveq2d 4685	. . . . . 6 |- (y = S -> ((cls` J)` (y \ {x})) = ((cls` J)` (S \ {x})))
27	26	eleq2d 1964	. . . . 5 |- (y = S -> (x e. ((cls` J)` (y \ {x})) <-> x e. ((cls` J)` (S \ {x}))))
28	27	abbidv 2008	. . . 4 |- (y = S -> {x | x e. ((cls` J)` (y \ {x}))} = {x | x e. ((cls` J)` (S \ {x}))})
29		eqid 1884	. . . 4 |- {<.y, z>. | (y e. ~PX /\ z = {x | x e. ((cls` J)` (y \ {x}))})} = {<.y, z>. | (y e. ~PX /\ z = {x | x e. ((cls` J)` (y \ {x}))})}
30	28, 29	fvopab4g 4742	. . 3 |- ((S e. ~PX /\ {x | x e. ((cls` J)` (S \ {x}))} e. _V) -> ({<.y, z>. | (y e. ~PX /\ z = {x | x e. ((cls` J)` (y \ {x}))})}` S) = {x | x e. ((cls` J)` (S \ {x}))})
31	14, 24, 30	syl11anc 524	. 2 |- ((J e. Top /\ S C_ X) -> ({<.y, z>. | (y e. ~PX /\ z = {x | x e. ((cls` J)` (y \ {x}))})}` S) = {x | x e. ((cls` J)` (S \ {x}))})
32	9, 31	eqtrd 1925	1 |- ((J e. Top /\ S C_ X) -> ((limPt` J)` S) = {x | x e. ((cls` J)` (S \ {x}))})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   \ cdif 2590   C_ wss 2593  ~Pcpw 3032  {csn 3044  U.cuni 3177  {copab 3395  ` cfv 3998  Topctop 8857  clsccl 8938  limPtclp 9016

This theorem is referenced by:  islp 9020  lpsscls 9021

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-top 8861  df-cld 8939  df-cls 8941  df-lp 9017

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