Theorem clsval2 7685

Description: Express closure in terms of interior.

Hypothesis

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

Assertion

Ref	Expression
clsval2	|- ((J e. Top /\ S (_ 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 7672	. . . . . . . . . . . . 13 |- ((J e. Top /\ z e. (Clsd` J)) -> (X \ z) e. J)
3		sseq1 2082	. . . . . . . . . . . . . . 15 |- (y = (X \ z) -> (y (_ (X \ S) <-> (X \ z) (_ (X \ S)))
4		eleq2 1535	. . . . . . . . . . . . . . . 16 |- (y = (X \ z) -> (x e. y <-> x e. (X \ z)))
5	4	negbid 611	. . . . . . . . . . . . . . 15 |- (y = (X \ z) -> (-. x e. y <-> -. x e. (X \ z)))
6	3, 5	imbi12d 626	. . . . . . . . . . . . . 14 |- (y = (X \ z) -> ((y (_ (X \ S) -> -. x e. y) <-> ((X \ z) (_ (X \ S) -> -. x e. (X \ z))))
7	6	rcla4v 1873	. . . . . . . . . . . . 13 |- ((X \ z) e. J -> (A.y e. J (y (_ (X \ S) -> -. x e. y) -> ((X \ z) (_ (X \ S) -> -. x e. (X \ z))))
8	2, 7	syl 10	. . . . . . . . . . . 12 |- ((J e. Top /\ z e. (Clsd` J)) -> (A.y e. J (y (_ (X \ S) -> -. x e. y) -> ((X \ z) (_ (X \ S) -> -. x e. (X \ z))))
9	8	adantrl 394	. . . . . . . . . . 11 |- ((J e. Top /\ (x e. X /\ z e. (Clsd` J))) -> (A.y e. J (y (_ (X \ S) -> -. x e. y) -> ((X \ z) (_ (X \ S) -> -. x e. (X \ z))))
10		sscon 2171	. . . . . . . . . . . . . 14 |- (S (_ z -> (X \ z) (_ (X \ S))
11	10	a1i 8	. . . . . . . . . . . . 13 |- (x e. X -> (S (_ z -> (X \ z) (_ (X \ S)))
12		neldif 2165	. . . . . . . . . . . . . 14 |- ((x e. X /\ -. x e. (X \ z)) -> x e. z)
13	12	ex 373	. . . . . . . . . . . . 13 |- (x e. X -> (-. x e. (X \ z) -> x e. z))
14	11, 13	imim12d 29	. . . . . . . . . . . 12 |- (x e. X -> (((X \ z) (_ (X \ S) -> -. x e. (X \ z)) -> (S (_ z -> x e. z)))
15	14	ad2antrl 406	. . . . . . . . . . 11 |- ((J e. Top /\ (x e. X /\ z e. (Clsd` J))) -> (((X \ z) (_ (X \ S) -> -. x e. (X \ z)) -> (S (_ z -> x e. z)))
16	9, 15	syld 27	. . . . . . . . . 10 |- ((J e. Top /\ (x e. X /\ z e. (Clsd` J))) -> (A.y e. J (y (_ (X \ S) -> -. x e. y) -> (S (_ z -> x e. z)))
17	16	exp32 377	. . . . . . . . 9 |- (J e. Top -> (x e. X -> (z e. (Clsd` J) -> (A.y e. J (y (_ (X \ S) -> -. x e. y) -> (S (_ z -> x e. z)))))
18	17	com34 36	. . . . . . . 8 |- (J e. Top -> (x e. X -> (A.y e. J (y (_ (X \ S) -> -. x e. y) -> (z e. (Clsd` J) -> (S (_ z -> x e. z)))))
19	18	imp3a 361	. . . . . . 7 |- (J e. Top -> ((x e. X /\ A.y e. J (y (_ (X \ S) -> -. x e. y)) -> (z e. (Clsd` J) -> (S (_ z -> x e. z))))
20	19	r19.21adv 1718	. . . . . 6 |- (J e. Top -> ((x e. X /\ A.y e. J (y (_ (X \ S) -> -. x e. y)) -> A.z e. (Clsd` J)(S (_ z -> x e. z)))
21	20	adantr 389	. . . . 5 |- ((J e. Top /\ S (_ X) -> ((x e. X /\ A.y e. J (y (_ (X \ S) -> -. x e. y)) -> A.z e. (Clsd` J)(S (_ z -> x e. z)))
22	1	topcld 7675	. . . . . . . . 9 |- (J e. Top -> X e. (Clsd` J))
23		sseq2 2083	. . . . . . . . . . 11 |- (z = X -> (S (_ z <-> S (_ X))
24		eleq2 1535	. . . . . . . . . . 11 |- (z = X -> (x e. z <-> x e. X))
25	23, 24	imbi12d 626	. . . . . . . . . 10 |- (z = X -> ((S (_ z -> x e. z) <-> (S (_ X -> x e. X)))
26	25	rcla4v 1873	. . . . . . . . 9 |- (X e. (Clsd` J) -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> (S (_ X -> x e. X)))
27	22, 26	syl 10	. . . . . . . 8 |- (J e. Top -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> (S (_ X -> x e. X)))
28	27	com23 32	. . . . . . 7 |- (J e. Top -> (S (_ X -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> x e. X)))
29	28	imp 350	. . . . . 6 |- ((J e. Top /\ S (_ X) -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> x e. X))
30	1	opncld 7674	. . . . . . . . . 10 |- ((J e. Top /\ y e. J) -> (X \ y) e. (Clsd` J))
31		sseq2 2083	. . . . . . . . . . . 12 |- (z = (X \ y) -> (S (_ z <-> S (_ (X \ y)))
32		eleq2 1535	. . . . . . . . . . . 12 |- (z = (X \ y) -> (x e. z <-> x e. (X \ y)))
33	31, 32	imbi12d 626	. . . . . . . . . . 11 |- (z = (X \ y) -> ((S (_ z -> x e. z) <-> (S (_ (X \ y) -> x e. (X \ y))))
34	33	rcla4v 1873	. . . . . . . . . 10 |- ((X \ y) e. (Clsd` J) -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> (S (_ (X \ y) -> x e. (X \ y))))
35	30, 34	syl 10	. . . . . . . . 9 |- ((J e. Top /\ y e. J) -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> (S (_ (X \ y) -> x e. (X \ y))))
36	35	adantlr 393	. . . . . . . 8 |- (((J e. Top /\ S (_ X) /\ y e. J) -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> (S (_ (X \ y) -> x e. (X \ y))))
37		ssconb 2170	. . . . . . . . . . . 12 |- ((y (_ X /\ S (_ X) -> (y (_ (X \ S) <-> S (_ (X \ y)))
38	37	biimpd 153	. . . . . . . . . . 11 |- ((y (_ X /\ S (_ X) -> (y (_ (X \ S) -> S (_ (X \ y)))
39	1	eltopss 7603	. . . . . . . . . . 11 |- ((J e. Top /\ y e. J) -> y (_ X)
40	38, 39	sylan 448	. . . . . . . . . 10 |- (((J e. Top /\ y e. J) /\ S (_ X) -> (y (_ (X \ S) -> S (_ (X \ y)))
41	40	an1rs 489	. . . . . . . . 9 |- (((J e. Top /\ S (_ X) /\ y e. J) -> (y (_ (X \ S) -> S (_ (X \ y)))
42		eldifn 2163	. . . . . . . . . 10 |- (x e. (X \ y) -> -. x e. y)
43	42	a1i 8	. . . . . . . . 9 |- (((J e. Top /\ S (_ X) /\ y e. J) -> (x e. (X \ y) -> -. x e. y))
44	41, 43	imim12d 29	. . . . . . . 8 |- (((J e. Top /\ S (_ X) /\ y e. J) -> ((S (_ (X \ y) -> x e. (X \ y)) -> (y (_ (X \ S) -> -. x e. y)))
45	36, 44	syld 27	. . . . . . 7 |- (((J e. Top /\ S (_ X) /\ y e. J) -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> (y (_ (X \ S) -> -. x e. y)))
46	45	r19.21adva 1719	. . . . . 6 |- ((J e. Top /\ S (_ X) -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> A.y e. J (y (_ (X \ S) -> -. x e. y)))
47	29, 46	jcad 600	. . . . 5 |- ((J e. Top /\ S (_ X) -> (A.z e. (Clsd` J)(S (_ z -> x e. z) -> (x e. X /\ A.y e. J (y (_ (X \ S) -> -. x e. y))))
48	21, 47	impbid 516	. . . 4 |- ((J e. Top /\ S (_ X) -> ((x e. X /\ A.y e. J (y (_ (X \ S) -> -. x e. y)) <-> A.z e. (Clsd` J)(S (_ z -> x e. z)))
49		eldif 2057	. . . . 5 |- (x e. (X \ U.{y e. J | y (_ (X \ S)}) <-> (x e. X /\ -. x e. U.{y e. J | y (_ (X \ S)}))
50		ralnex 1653	. . . . . . 7 |- (A.y e. J -. (x e. y /\ y (_ (X \ S)) <-> -. E.y e. J (x e. y /\ y (_ (X \ S)))
51		imnan 242	. . . . . . . . 9 |- ((y (_ (X \ S) -> -. x e. y) <-> -. (y (_ (X \ S) /\ x e. y))
52		ancom 435	. . . . . . . . . 10 |- ((y (_ (X \ S) /\ x e. y) <-> (x e. y /\ y (_ (X \ S)))
53	52	negbii 187	. . . . . . . . 9 |- (-. (y (_ (X \ S) /\ x e. y) <-> -. (x e. y /\ y (_ (X \ S)))
54	51, 53	bitr 173	. . . . . . . 8 |- ((y (_ (X \ S) -> -. x e. y) <-> -. (x e. y /\ y (_ (X \ S)))
55	54	ralbii 1667	. . . . . . 7 |- (A.y e.