Theorem cldbnd 32321

Description: A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)

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 20868	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( A e. ( Clsd ` J ) <-> ( ( cls ` J ) ` A ) = A ) )
3		eqimss 3657	. . . 4 |- ( ( ( cls ` J ) ` A ) = A -> ( ( cls ` J ) ` A ) C_ A )
4	2, 3	syl6bi 243	. . 3 |- ( ( J e. Top /\ A C_ X ) -> ( A e. ( Clsd ` J ) -> ( ( cls ` J ) ` A ) C_ A ) )
5		ssinss1 3841	. . 3 |- ( ( ( cls ` J ) ` A ) C_ A -> ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) C_ A )
6	4, 5	syl6 35	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( A e. ( Clsd ` J ) -> ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) C_ A ) )
7		sslin 3839	. . . . . 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 482	. . . . 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 3805	. . . . . 6 |- ( ( X \ A ) i^i A ) = ( A i^i ( X \ A ) )
10		disjdif 4040	. . . . . 6 |- ( A i^i ( X \ A ) ) = (/)
11	9, 10	eqtri 2644	. . . . 5 |- ( ( X \ A ) i^i A ) = (/)
12		sseq0 3975	. . . . 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 ) ) ) ) = (/) )
13	8, 11, 12	sylancl 694	. . . 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 ) ) ) ) = (/) )
14	13	ex 450	. . 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 ) ) ) ) = (/) ) )
15		incom 3805	. . . . . . . 8 |- ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) = ( ( ( cls ` J ) ` ( X \ A ) ) i^i ( ( cls ` J ) ` A ) )
16		dfss4 3858	. . . . . . . . . . 11 |- ( A C_ X <-> ( X \ ( X \ A ) ) = A )
17		fveq2 6191	. . . . . . . . . . . 12 |- ( ( X \ ( X \ A ) ) = A -> ( ( cls ` J ) ` ( X \ ( X \ A ) ) ) = ( ( cls ` J ) ` A ) )
18	17	eqcomd 2628	. . . . . . . . . . 11 |- ( ( X \ ( X \ A ) ) = A -> ( ( cls ` J ) ` A ) = ( ( cls ` J ) ` ( X \ ( X \ A ) ) ) )
19	16, 18	sylbi 207	. . . . . . . . . 10 |- ( A C_ X -> ( ( cls ` J ) ` A ) = ( ( cls ` J ) ` ( X \ ( X \ A ) ) ) )
20	19	adantl 482	. . . . . . . . 9 |- ( ( J e. Top /\ A C_ X ) -> ( ( cls ` J ) ` A ) = ( ( cls ` J ) ` ( X \ ( X \ A ) ) ) )
21	20	ineq2d 3814	. . . . . . . 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	15, 21	syl5eq 2668	. . . . . . 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 ) ) ) ) )
23	22	ineq2d 3814	. . . . . 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 ) ) ) ) ) )
24	23	eqeq1d 2624	. . . . 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 ) ) ) ) ) = (/) ) )
25		difss 3737	. . . . . . 7 |- ( X \ A ) C_ X
26	1	opnbnd 32320	. . . . . . 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 ) ) ) ) ) = (/) ) )
27	25, 26	mpan2 707	. . . . . 6 |- ( J e. Top -> ( ( X \ A ) e. J <-> ( ( X \ A ) i^i ( ( ( cls ` J ) ` ( X \ A ) ) i^i ( ( cls ` J ) ` ( X \ ( X \ A ) ) ) ) ) = (/) ) )
28	27	adantr 481	. . . . 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 ) ) ) ) ) = (/) ) )
29	24, 28	bitr4d 271	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( X \ A ) i^i ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) ) = (/) <-> ( X \ A ) e. J ) )
30	1	opncld 20837	. . . . . . 7 |- ( ( J e. Top /\ ( X \ A ) e. J ) -> ( X \ ( X \ A ) ) e. ( Clsd ` J ) )
31	30	ex 450	. . . . . 6 |- ( J e. Top -> ( ( X \ A ) e. J -> ( X \ ( X \ A ) ) e. ( Clsd ` J ) ) )
32	31	adantr 481	. . . . 5 |- ( ( J e. Top /\ A C_ X ) -> ( ( X \ A ) e. J -> ( X \ ( X \ A ) ) e. ( Clsd ` J ) ) )
33		eleq1 2689	. . . . . . 7 |- ( ( X \ ( X \ A ) ) = A -> ( ( X \ ( X \ A ) ) e. ( Clsd ` J ) <-> A e. ( Clsd ` J ) ) )
34	16, 33	sylbi 207	. . . . . 6 |- ( A C_ X -> ( ( X \ ( X \ A ) ) e. ( Clsd ` J ) <-> A e. ( Clsd ` J ) ) )
35	34	adantl 482	. . . . 5 |- ( ( J e. Top /\ A C_ X ) -> ( ( X \ ( X \ A ) ) e. ( Clsd ` J ) <-> A e. ( Clsd ` J ) ) )
36	32, 35	sylibd 229	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( X \ A ) e. J -> A e. ( Clsd ` J ) ) )
37	29, 36	sylbid 230	. . 3 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( X \ A ) i^i ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) ) = (/) -> A e. ( Clsd ` J ) ) )
38	14, 37	syld 47	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) C_ A -> A e. ( Clsd ` J ) ) )
39	6, 38	impbid 202	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 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   ` cfv 5888   Topctop 20698   Clsdccld 20820   clsccl 20822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by: (None)