Theorem lpval 7693

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 (_ 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 7692	. . . . 5 |- (J e. Top -> (limPt` J) = {<.y, z>. | (y (_ X /\ z = {x | x e. ((cls` J)` (y \ {x}))})})
3	2	adantr 389	. . . 4 |- ((J e. Top /\ S (_ X) -> (limPt` J) = {<.y, z>. | (y (_ X /\ z = {x | x e. ((cls` J)` (y \ {x}))})})
4		visset 1809	. . . . . . 7 |- y e. V
5	4	elpw 2400	. . . . . 6 |- (y e. P~X <-> y (_ X)
6	5	anbi1i 481	. . . . 5 |- ((y e. P~X /\ z = {x | x e. ((cls` J)` (y \ {x}))}) <-> (y (_ X /\ z = {x | x e. ((cls` J)` (y \ {x}))}))
7	6	opabbii 2666	. . . 4 |- {<.y, z>. | (y e. P~X /\ z = {x | x e. ((cls` J)` (y \ {x}))})} = {<.y, z>. | (y (_ X /\ z = {x | x e. ((cls` J)` (y \ {x}))})}
8	3, 7	syl6eqr 1522	. . 3 |- ((J e. Top /\ S (_ X) -> (limPt` J) = {<.y, z>. | (y e. P~X /\ z = {x | x e. ((cls` J)` (y \ {x}))})})
9	8	fveq1d 3717	. 2 |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) = ({<.y, z>. | (y e. P~X /\ z = {x | x e. ((cls` J)` (y \ {x}))})}` S))
10		difeq1 2149	. . . . . . 7 |- (y = S -> (y \ {x}) = (S \ {x}))
11	10	fveq2d 3719	. . . . . 6 |- (y = S -> ((cls` J)` (y \ {x})) = ((cls` J)` (S \ {x})))
12	11	eleq2d 1538	. . . . 5 |- (y = S -> (x e. ((cls` J)` (y \ {x})) <-> x e. ((cls` J)` (S \ {x}))))
13	12	abbidv 1574	. . . 4 |- (y = S -> {x | x e. ((cls` J)` (y \ {x}))} = {x | x e. ((cls` J)` (S \ {x}))})
14		eqid 1473	. . . 4 |- {<.y, z>. | (y e. P~X /\ z = {x | x e. ((cls` J)` (y \ {x}))})} = {<.y, z>. | (y e. P~X /\ z = {x | x e. ((cls` J)` (y \ {x}))})}
15	13, 14	fvopab4g 3770	. . 3 |- ((S e. P~X /\ {x | x e. ((cls` J)` (S \ {x}))} e. V) -> ({<.y, z>. | (y e. P~X /\ z = {x | x e. ((cls` J)` (y \ {x}))})}` S) = {x | x e. ((cls` J)` (S \ {x}))})
16		elpw2g 2722	. . . . 5 |- (X e. V -> (S e. P~X <-> S (_ X))
17	16	biimpar 417	. . . 4 |- ((X e. V /\ S (_ X) -> S e. P~X)
18		uniexg 2866	. . . . 5 |- (J e. Top -> U.J e. V)
19	18, 1	syl5eqel 1549	. . . 4 |- (J e. Top -> X e. V)
20	17, 19	sylan 448	. . 3 |- ((J e. Top /\ S (_ X) -> S e. P~X)
21		difss 2163	. . . . . . . 8 |- (S \ {x}) (_ S
22	1	clsss 7637	. . . . . . . 8 |- ((J e. Top /\ S (_ X /\ (S \ {x}) (_ S) -> ((cls` J)` (S \ {x})) (_ ((cls` J)` S))
23	21, 22	mp3an3 903	. . . . . . 7 |- ((J e. Top /\ S (_ X) -> ((cls` J)` (S \ {x})) (_ ((cls` J)` S))
24	23	sseld 2063	. . . . . 6 |- ((J e. Top /\ S (_ X) -> (x e. ((cls` J)` (S \ {x})) -> x e. ((cls` J)` S)))
25	24	19.21aiv 1284	. . . . 5 |- ((J e. Top /\ S (_ X) -> A.x(x e. ((cls` J)` (S \ {x})) -> x e. ((cls` J)` S)))
26		ss2ab 2112	. . . . 5 |- ({x | x e. ((cls` J)` (S \ {x}))} (_ {x | x e. ((cls` J)` S)} <-> A.x(x e. ((cls` J)` (S \ {x})) -> x e. ((cls` J)` S)))
27	25, 26	sylibr 200	. . . 4 |- ((J e. Top /\ S (_ X) -> {x | x e. ((cls` J)` (S \ {x}))} (_ {x | x e. ((cls` J)` S)})
28		abid2 1577	. . . . . 6 |- {x | x e. ((cls` J)` S)} = ((cls` J)` S)
29		fvex 3723	. . . . . 6 |- ((cls` J)` S) e. V
30	28, 29	eqeltr 1541	. . . . 5 |- {x | x e. ((cls` J)` S)} e. V
31	30	ssex 2714	. . . 4 |- ({x | x e. ((cls` J)` (S \ {x}))} (_ {x | x e. ((cls` J)` S)} -> {x | x e. ((cls` J)` (S \ {x}))} e. V)
32	27, 31	syl 10	. . 3 |- ((J e. Top /\ S (_ X) -> {x | x e. ((cls` J)` (S \ {x}))} e. V)
33	15, 20, 32	sylanc 471	. 2 |- ((J e. Top /\ S (_ X) -> ({<.y, z>. | (y e. P~X /\ z = {x | x e. ((cls` J)` (y \ {x}))})}` S) = {x | x e. ((cls` J)` (S \ {x}))})
34	9, 33	eqtrd 1504	1 |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) = {x | x e. ((cls` J)` (S \ {x}))})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  {cab 1461  Vcvv 1807   \ cdif 2040   (_ wss 2043  P~cpw 2397  {csn 2405  U.cuni 2498  {copab 2661  ` cfv 3177  Topctop 7538  clsccl 7612  limPtclp 7690

This theorem is referenced by:  islp 7694  lpsscls 7695

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-top 7542  df-cld 7613  df-cls 7615  df-lp 7691

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