Theorem clsval2 8961

Description: Express closure in terms of interior.

Hypothesis

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

Assertion

Ref	Expression
clsval2	|- ((J e. Top /\ S C_ X) -> ((cls` J)` S) = (X \ ((int` J)` (X \ S))))

Proof of Theorem clsval2

Step	Hyp	Ref	Expression
1		clscld.1	. . . . . . . . . . . . . 14 |- X = U.J
2	1	cldopn 8948	. . . . . . . . . . . . 13 |- ((J e. Top /\ z e. (Clsd` J)) -> (X \ z) e. J)
3		sseq1 2637	. . . . . . . . . . . . . . 15 |- (y = (X \ z) -> (y C_ (X \ S) <-> (X \ z) C_ (X \ S)))
4		eleq2 1958	. . . . . . . . . . . . . . . 16 |- (y = (X \ z) -> (x e. y <-> x e. (X \ z)))
5	4	notbid 673	. . . . . . . . . . . . . . 15 |- (y = (X \ z) -> (-. x e. y <-> -. x e. (X \ z)))
6	3, 5	imbi12d 688	. . . . . . . . . . . . . 14 |- (y = (X \ z) -> ((y C_ (X \ S) -> -. x e. y) <-> ((X \ z) C_ (X \ S) -> -. x e. (X \ z))))
7	6	rcla4v 2376	. . . . . . . . . . . . 13 |- ((X \ z) e. J -> (A.y e. J (y C_ (X \ S) -> -. x e. y) -> ((X \ z) C_ (X \ S) -> -. x e. (X \ z))))
8	2, 7	syl 12	. . . . . . . . . . . 12 |- ((J e. Top /\ z e. (Clsd` J)) -> (A.y e. J (y C_ (X \ S) -> -. x e. y) -> ((X \ z) C_ (X \ S) -> -. x e. (X \ z))))
9	8	adantrl 430	. . . . . . . . . . 11 |- ((J e. Top /\ (x e. X /\ z e. (Clsd` J))) -> (A.y e. J (y C_ (X \ S) -> -. x e. y) -> ((X \ z) C_ (X \ S) -> -. x e. (X \ z))))
10		sscon 2739	. . . . . . . . . . . . . 14 |- (S C_ z -> (X \ z) C_ (X \ S))
11	10	a1i 8	. . . . . . . . . . . . 13 |- (x e. X -> (S C_ z -> (X \ z) C_ (X \ S)))
12		neldif 2733	. . . . . . . . . . . . . 14 |- ((x e. X /\ -. x e. (X \ z)) -> x e. z)
13	12	ex 402	. . . . . . . . . . . . 13 |- (x e. X -> (-. x e. (X \ z) -> x e. z))
14	11, 13	imim12d 69	. . . . . . . . . . . 12 |- (x e. X -> (((X \ z) C_ (X \ S) -> -. x e. (X \ z)) -> (S C_ z -> x e. z)))
15	14	ad2antrl 442	. . . . . . . . . . 11 |- ((J e. Top /\ (x e. X /\ z e. (Clsd` J))) -> (((X \ z) C_ (X \ S) -> -. x e. (X \ z)) -> (S C_ z -> x e. z)))
16	9, 15	syld 30	. . . . . . . . . 10 |- ((J e. Top /\ (x e. X /\ z e. (Clsd` J))) -> (A.y e. J (y C_ (X \ S) -> -. x e. y) -> (S C_ z -> x e. z)))
17	16	exp32 408	. . . . . . . . 9 |- (J e. Top -> (x e. X -> (z e. (Clsd` J) -> (A.y e. J (y C_ (X \ S) -> -. x e. y) -> (S C_ z -> x e. z)))))
18	17	com34 40	. . . . . . . 8 |- (J e. Top -> (x e. X -> (A.y e. J (y C_ (X \ S) -> -. x e. y) -> (z e. (Clsd` J) -> (S C_ z -> x e. z)))))
19	18	imp3a 388	. . . . . . 7 |- (J e. Top -> ((x e. X /\ A.y e. J (y C_ (X \ S) -> -. x e. y)) -> (z e. (Clsd` J) -> (S C_ z -> x e. z))))
20	19	r19.21adv 2181	. . . . . 6 |- (J e. Top -> ((x e. X /\ A.y e. J (y C_ (X \ S) -> -. x e. y)) -> A.z e. (Clsd` J)(S C_ z -> x e. z)))
21	20	adantr 425	. . . . 5 |- ((J e. Top /\ S C_ X) -> ((x e. X /\ A.y e. J (y C_ (X \ S) -> -. x e. y)) -> A.z e. (Clsd` J)(S C_ z -> x e. z)))
22	1	topcld 8951	. . . . . . . . 9 |- (J e. Top -> X e. (Clsd` J))
23		sseq2 2639	. . . . . . . . . . 11 |- (z = X -> (S C_ z <-> S C_ X))
24		eleq2 1958	. . . . . . . . . . 11 |- (z = X -> (x e. z <-> x e. X))
25	23, 24	imbi12d 688	. . . . . . . . . 10 |- (z = X -> ((S C_ z -> x e. z) <-> (S C_ X -> x e. X)))
26	25	rcla4v 2376	. . . . . . . . 9 |- (X e. (Clsd` J) -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> (S C_ X -> x e. X)))
27	22, 26	syl 12	. . . . . . . 8 |- (J e. Top -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> (S C_ X -> x e. X)))
28	27	com23 36	. . . . . . 7 |- (J e. Top -> (S C_ X -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> x e. X)))
29	28	imp 377	. . . . . 6 |- ((J e. Top /\ S C_ X) -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> x e. X))
30	1	opncld 8950	. . . . . . . . . 10 |- ((J e. Top /\ y e. J) -> (X \ y) e. (Clsd` J))
31		sseq2 2639	. . . . . . . . . . . 12 |- (z = (X \ y) -> (S C_ z <-> S C_ (X \ y)))
32		eleq2 1958	. . . . . . . . . . . 12 |- (z = (X \ y) -> (x e. z <-> x e. (X \ y)))
33	31, 32	imbi12d 688	. . . . . . . . . . 11 |- (z = (X \ y) -> ((S C_ z -> x e. z) <-> (S C_ (X \ y) -> x e. (X \ y))))
34	33	rcla4v 2376	. . . . . . . . . 10 |- ((X \ y) e. (Clsd` J) -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> (S C_ (X \ y) -> x e. (X \ y))))
35	30, 34	syl 12	. . . . . . . . 9 |- ((J e. Top /\ y e. J) -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> (S C_ (X \ y) -> x e. (X \ y))))
36	35	adantlr 429	. . . . . . . 8 |- (((J e. Top /\ S C_ X) /\ y e. J) -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> (S C_ (X \ y) -> x e. (X \ y))))
37		ssconb 2738	. . . . . . . . . . . 12 |- ((y C_ X /\ S C_ X) -> (y C_ (X \ S) <-> S C_ (X \ y)))
38	37	biimpd 170	. . . . . . . . . . 11 |- ((y C_ X /\ S C_ X) -> (y C_ (X \ S) -> S C_ (X \ y)))
39	1	eltopss 8872	. . . . . . . . . . 11 |- ((J e. Top /\ y e. J) -> y C_ X)
40	38, 39	sylan 497	. . . . . . . . . 10 |- (((J e. Top /\ y e. J) /\ S C_ X) -> (y C_ (X \ S) -> S C_ (X \ y)))
41	40	an1rs 547	. . . . . . . . 9 |- (((J e. Top /\ S C_ X) /\ y e. J) -> (y C_ (X \ S) -> S C_ (X \ y)))
42		eldifn 2731	. . . . . . . . . 10 |- (x e. (X \ y) -> -. x e. y)
43	42	a1i 8	. . . . . . . . 9 |- (((J e. Top /\ S C_ X) /\ y e. J) -> (x e. (X \ y) -> -. x e. y))
44	41, 43	imim12d 69	. . . . . . . 8 |- (((J e. Top /\ S C_ X) /\ y e. J) -> ((S C_ (X \ y) -> x e. (X \ y)) -> (y C_ (X \ S) -> -. x e. y)))
45	36, 44	syld 30	. . . . . . 7 |- (((J e. Top /\ S C_ X) /\ y e. J) -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> (y C_ (X \ S) -> -. x e. y)))
46	45	r19.21adva 2182	. . . . . 6 |- ((J e. Top /\ S C_ X) -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> A.y e. J (y C_ (X \ S) -> -. x e. y)))
47	29, 46	jcad 661	. . . . 5 |- ((J e. Top /\ S C_ X) -> (A.z e. (Clsd` J)(S C_ z -> x e. z) -> (x e. X /\ A.y e. J (y C_ (X \ S) -> -. x e. y))))
48	21, 47	impbid 574	. . . 4 |- ((J e. Top /\ S C_ X) -> ((x e. X /\ A.y e. J (y C_ (X \ S) -> -. x e. y)) <-> A.z e. (Clsd` J)(S C_ z -> x e. z)))
49		eldif 2609	. . . . 5 |- (x e. (X \ U.{y e. J | y C_ (X \ S)}) <-> (x e. X /\ -. x e. U.{y e. J | y C_ (X \ S)}))
50		ralnex 2113	. . . . . . 7 |- (A.y e. J -. (x e. y /\ y C_ (X \ S)) <-> -. E.y e. J (x e. y /\ y C_ (X \ S)))
51		imnan 261	. . . . . . . . 9 |- ((y C_ (X \ S) -> -. x e. y) <-> -. (y C_ (X \ S) /\ x e. y))
52		ancom 482	. . . . . . . . . 10 |- ((y C_ (X \ S) /\ x e. y) <-> (x e. y /\ y C_ (X \ S)))
53	52	notbii 204	. . . . . . . . 9 |- (-. (y C_ (X \ S) /\ x e. y) <-> -. (x e. y /\ y C_ (X \ S)))
54	51, 53	bitri 190	. . . . . . . 8 |- ((y C_ (X \ S) -> -. x e. y) <-> -. (x e. y /\ y C_ (X \ S)))
55	54	ralbii 2127	. . . . . . 7 |- (A.y e. J (y C_ (X \ S) -> -. x e. y) <-> A.y e. J -. (x e. y /\ y C_ (X \ S)))
56		elunirab 3190	. . . . . . . 8 |- (x e. U.{y e. J | y C_ (X \ S)} <-> E.y e. J (x e. y /\ y C_ (X \ S)))
57	56	notbii 204	. . . . . . 7 |- (-. x e. U.{y e. J | y C_ (X \ S)} <-> -. E.y e. J (x e. y /\ y C_ (X \ S)))
58	50, 55, 57	3bitr4ri 201	. . . . . 6 |- (-. x e. U.{y e. J | y C_ (X \ S)} <-> A.y e. J (y C_ (X \ S) -> -. x e. y))
59	58	anbi2i 538	. . . . 5 |- ((x e. X /\ -. x e. U.{y e. J | y C_ (X \ S)}) <-> (x e. X /\ A.y e. J (y C_ (X \ S) -> -. x e. y)))
60	49, 59	bitri 190	. . . 4 |- (x e. (X \ U.{y e. J | y C_ (X \ S)}) <-> (x e. X /\ A.y e. J (y C_ (X \ S) -> -. x e. y)))
61		visset 2295	. . . . 5 |- x e. _V
62	61	elintrab 3228	. . . 4 |- (x e. |^|{z e. (Clsd` J) | S C_ z} <-> A.z e. (Clsd` J)(S C_ z -> x e. z))
63	48, 60, 62	3bitr4g 614	. . 3 |- ((J e. Top /\ S C_ X) -> (x e. (X \ U.{y e. J | y C_ (X \ S)}) <-> x e. |^|{z e. (Clsd` J) | S C_ z}))
64	63	eqrdv 1882	. 2 |- ((J e. Top /\ S C_ X) -> (X \ U.{y e. J | y C_ (X \ S)}) = |^|{z e. (Clsd` J) | S C_ z})
65	1	ntrval 8952	. . . 4 |- ((J e. Top /\ (X \ S) C_ X) -> ((int` J)` (X \ S)) = U.{y e. J | y C_ (X \ S)})
66		difss 2735	. . . . 5 |- (X \ S) C_ X
67	66	a1i 8	. . . 4 |- (S C_ X -> (X \ S) C_ X)
68	65, 67	sylan2 500	. . 3 |- ((J e. Top /\ S C_ X) -> ((int` J)` (X \ S)) = U.{y e. J | y C_ (X \ S)})
69	68	difeq2d 2726	. 2 |- ((J e. Top /\ S C_ X) -> (X \ ((int` J)` (X \ S))) = (X \ U.{y e. J | y C_ (X \ S)}))
70	1	clsval 8953	. 2 |- ((J e. Top /\ S C_ X) -> ((cls` J)` S) = |^|{z e. (Clsd` J) | S C_ z})
71	64, 69, 70	3eqtr4rd 1939	1 |- ((J e. Top /\ S C_ X) -> ((cls` J)` S) = (X \ ((int` J)` (X \ S))))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  {crab 2108   \ cdif 2590   C_ wss 2593  U.cuni 3177  |^|cint 3214  ` cfv 3998  Topctop 8857  Clsdccld 8936  intcnt 8937  clsccl 8938

This theorem is referenced by:  ntrval2 8962  cmclsopn 8969  clscmp 15407

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-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-ntr 8940  df-cls 8941

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