Theorem lpval 7743

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 7742	. . . . 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 1813	. . . . . . 7 |- y e. V
5	4	elpw 2404	. . . . . 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 2671	. . . 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 1525	. . 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 3726	. 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 2153	. . . . . . 7 |- (y = S -> (y \ {x}) = (S \ {x}))
11	10	fveq2d 3728	. . . . . 6 |- (y = S -> ((cls` J)` (y \ {x})) = ((cls` J)` (S \ {x})))
12	11	eleq2d 1541	. . . . 5 |- (y = S -> (x e. ((cls` J)` (y \ {x})) <-> x e. ((cls` J)` (S \ {x}))))
13	12	abbidv 1577	. . . 4 |- (y = S -> {x | x e. ((cls` J)` (y \ {x}))} = {x | x e. ((cls` J)` (S \ {x}))})
14		eqid 1475	. . . 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 3779	. . 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 2727	. . . . 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 2871	. . . . 5 |- (J e. Top -> U.J e. V)
19	18, 1	syl5eqel 1552	. . . 4 |- (J e. Top -> X e. V)
20	17, 19	sylan 448	. . 3 |- ((J e. Top /\ S (_ X) -> S e. P~X)
21		difss 2167	. . . . . . . 8 |- (S \ {x}) (_ S
22	1	clsss 7687	. . . . . . . 8 |- ((J e. Top /\ S (_ X /\ (S \ {x}) (_ S) -> ((cls` J)` (S \ {x})) (_ ((cls` J)` S))
23	21, 22	mp3an3 905	. . . . . . 7 |- ((J e. Top /\ S (_ X) -> ((cls` J)` (S \ {x})) (_ ((cls` J)` S))
24	23	sseld 2067	. . . . . 6 |- ((J e. Top /\ S (_ X) -> (x e. ((cls` J)` (S \ {x})) -> x e. ((cls` J)` S)))
25	24	19.21aiv 1286	. . . . 5 |- ((J e. Top /\ S (_ X) -> A.x(x e. ((cls` J)` (S \ {x})) -> x e. ((cls` J)` S)))
26		ss2ab 2116	. . . . 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 1580	. . . . . 6 |- {x | x e. ((cls` J)` S)} = ((cls` J)` S)
29		fvex 3732	. . . . . 6 |- ((cls` J)` S) e. V
30	28, 29	eqeltr 1544	. . . . 5 |- {x | x e. ((cls` J)` S)} e. V
31	30	ssex 2719	. . . 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 1507	1 |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) = {x | x e. ((cls` J)` (S \ {x}))})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044   (_ wss 2047  P~cpw 2401  {csn 2409  U.cuni 2503  {copab 2666  ` cfv 3182  Topctop 7588  clsccl 7662  limPtclp 7740

This theorem is referenced by:  islp 7744  lpsscls 7745

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-top 7592  df-cld 7663  df-cls 7665  df-lp 7741

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