Theorem cldbnd 15410

Description: A set is closed iff it contains its boundary.

Hypothesis

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

Assertion

Ref	Expression
cldbnd	|- ((J e. Top /\ A C_ X) -> (A e. (Clsd` J) <-> (((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A))

Proof of Theorem cldbnd

Step	Hyp	Ref	Expression
1		opnbnd.1	. . . . 5 |- X = U.J
2	1	iscld3 8971	. . . 4 |- ((J e. Top /\ A C_ X) -> (A e. (Clsd` J) <-> ((cls` J)` A) = A))
3		eqimss 2665	. . . 4 |- (((cls` J)` A) = A -> ((cls` J)` A) C_ A)
4	2, 3	syl6bi 231	. . 3 |- ((J e. Top /\ A C_ X) -> (A e. (Clsd` J) -> ((cls` J)` A) C_ A))
5		ssinss1 2821	. . 3 |- (((cls` J)` A) C_ A -> (((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A)
6	4, 5	syl6 25	. 2 |- ((J e. Top /\ A C_ X) -> (A e. (Clsd` J) -> (((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A))
7		sslin 2819	. . . . . 6 |- ((((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A -> ((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) C_ ((X \ A) i^i A))
8	7	adantl 424	. . . . 5 |- (((J e. Top /\ A C_ X) /\ (((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A) -> ((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) C_ ((X \ A) i^i A))
9		incom 2787	. . . . . . 7 |- ((X \ A) i^i A) = (A i^i (X \ A))
10		difdisj 2945	. . . . . . 7 |- (A i^i (X \ A)) = (/)
11	9, 10	eqtri 1908	. . . . . 6 |- ((X \ A) i^i A) = (/)
12	11	a1i 8	. . . . 5 |- (((J e. Top /\ A C_ X) /\ (((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A) -> ((X \ A) i^i A) = (/))
13		sseq0 2903	. . . . 5 |- ((((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) C_ ((X \ A) i^i A) /\ ((X \ A) i^i A) = (/)) -> ((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/))
14	8, 12, 13	syl11anc 524	. . . 4 |- (((J e. Top /\ A C_ X) /\ (((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A) -> ((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/))
15	14	ex 402	. . 3 |- ((J e. Top /\ A C_ X) -> ((((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A -> ((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/)))
16		dfss4 2827	. . . . . . . . . . 11 |- (A C_ X <-> (X \ (X \ A)) = A)
17		fveq2 4681	. . . . . . . . . . . 12 |- ((X \ (X \ A)) = A -> ((cls` J)` (X \ (X \ A))) = ((cls` J)` A))
18	17	eqcomd 1889	. . . . . . . . . . 11 |- ((X \ (X \ A)) = A -> ((cls` J)` A) = ((cls` J)` (X \ (X \ A))))
19	16, 18	sylbi 216	. . . . . . . . . 10 |- (A C_ X -> ((cls` J)` A) = ((cls` J)` (X \ (X \ A))))
20	19	adantl 424	. . . . . . . . 9 |- ((J e. Top /\ A C_ X) -> ((cls` J)` A) = ((cls` J)` (X \ (X \ A))))
21	20	ineq2d 2796	. . . . . . . 8 |- ((J e. Top /\ A C_ X) -> (((cls` J)` (X \ A)) i^i ((cls` J)` A)) = (((cls` J)` (X \ A)) i^i ((cls` J)` (X \ (X \ A)))))
22		incom 2787	. . . . . . . 8 |- (((cls` J)` A) i^i ((cls` J)` (X \ A))) = (((cls` J)` (X \ A)) i^i ((cls` J)` A))
23	21, 22	syl5eq 1940	. . . . . . 7 |- ((J e. Top /\ A C_ X) -> (((cls` J)` A) i^i ((cls` J)` (X \ A))) = (((cls` J)` (X \ A)) i^i ((cls` J)` (X \ (X \ A)))))
24	23	ineq2d 2796	. . . . . 6 |- ((J e. Top /\ A C_ X) -> ((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = ((X \ A) i^i (((cls` J)` (X \ A)) i^i ((cls` J)` (X \ (X \ A))))))
25	24	eqeq1d 1892	. . . . 5 |- ((J e. Top /\ A C_ X) -> (((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/) <-> ((X \ A) i^i (((cls` J)` (X \ A)) i^i ((cls` J)` (X \ (X \ A))))) = (/)))
26		difss 2735	. . . . . . 7 |- (X \ A) C_ X
27	1	opnbnd 15409	. . . . . . 7 |- ((J e. Top /\ (X \ A) C_ X) -> ((X \ A) e. J <-> ((X \ A) i^i (((cls` J)` (X \ A)) i^i ((cls` J)` (X \ (X \ A))))) = (/)))
28	26, 27	mpan2 760	. . . . . 6 |- (J e. Top -> ((X \ A) e. J <-> ((X \ A) i^i (((cls` J)` (X \ A)) i^i ((cls` J)` (X \ (X \ A))))) = (/)))
29	28	adantr 425	. . . . 5 |- ((J e. Top /\ A C_ X) -> ((X \ A) e. J <-> ((X \ A) i^i (((cls` J)` (X \ A)) i^i ((cls` J)` (X \ (X \ A))))) = (/)))
30	25, 29	bitr4d 590	. . . 4 |- ((J e. Top /\ A C_ X) -> (((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/) <-> (X \ A) e. J))
31	1	opncld 8950	. . . . . . 7 |- ((J e. Top /\ (X \ A) e. J) -> (X \ (X \ A)) e. (Clsd` J))
32	31	ex 402	. . . . . 6 |- (J e. Top -> ((X \ A) e. J -> (X \ (X \ A)) e. (Clsd` J)))
33	32	adantr 425	. . . . 5 |- ((J e. Top /\ A C_ X) -> ((X \ A) e. J -> (X \ (X \ A)) e. (Clsd` J)))
34		eleq1 1957	. . . . . . 7 |- ((X \ (X \ A)) = A -> ((X \ (X \ A)) e. (Clsd` J) <-> A e. (Clsd` J)))
35	16, 34	sylbi 216	. . . . . 6 |- (A C_ X -> ((X \ (X \ A)) e. (Clsd` J) <-> A e. (Clsd` J)))
36	35	adantl 424	. . . . 5 |- ((J e. Top /\ A C_ X) -> ((X \ (X \ A)) e. (Clsd` J) <-> A e. (Clsd` J)))
37	33, 36	sylibd 219	. . . 4 |- ((J e. Top /\ A C_ X) -> ((X \ A) e. J -> A e. (Clsd` J)))
38	30, 37	sylbid 220	. . 3 |- ((J e. Top /\ A C_ X) -> (((X \ A) i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/) -> A e. (Clsd` J)))
39	15, 38	syld 30	. 2 |- ((J e. Top /\ A C_ X) -> ((((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A -> A e. (Clsd` J)))
40	6, 39	impbid 574	1 |- ((J e. Top /\ A C_ X) -> (A e. (Clsd` J) <-> (((cls` J)` A) i^i ((cls` J)` (X \ A))) C_ A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   i^i cin 2592   C_ wss 2593  (/)c0 2875  U.cuni 3177  ` cfv 3998  Topctop 8857  Clsdccld 8936  clsccl 8938

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-sbc 2454  df-csb 2541  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-iun 3257  df-iin 3258  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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