Theorem opnbnd 30070

Description: A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)

Hypothesis

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

Assertion

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

Proof of Theorem opnbnd

Step	Hyp	Ref	Expression
1		disjdif 3905	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/)
2	1	a1i 11	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) )
3		ineq1 3698	. . . . 5 |- ( ( ( int ` J ) ` A ) = A -> ( ( ( int ` J ) ` A ) i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) )
4	3	eqeq1d 2469	. . . 4 |- ( ( ( int ` J ) ` A ) = A -> ( ( ( ( int ` J ) ` A ) i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) )
5	2, 4	syl5ibcom 220	. . 3 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( int ` J ) ` A ) = A -> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) )
6		opnbnd.1	. . . . . . 7 |- X = U. J
7	6	ntrss2 19426	. . . . . 6 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) C_ A )
8	7	adantr 465	. . . . 5 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> ( ( int ` J ) ` A ) C_ A )
9		inssdif0 3900	. . . . . 6 |- ( ( A i^i ( ( cls ` J ) ` A ) ) C_ ( ( int ` J ) ` A ) <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) )
10	6	sscls 19425	. . . . . . . . . 10 |- ( ( J e. Top /\ A C_ X ) -> A C_ ( ( cls ` J ) ` A ) )
11		df-ss 3495	. . . . . . . . . 10 |- ( A C_ ( ( cls ` J ) ` A ) <-> ( A i^i ( ( cls ` J ) ` A ) ) = A )
12	10, 11	sylib 196	. . . . . . . . 9 |- ( ( J e. Top /\ A C_ X ) -> ( A i^i ( ( cls ` J ) ` A ) ) = A )
13	12	eqcomd 2475	. . . . . . . 8 |- ( ( J e. Top /\ A C_ X ) -> A = ( A i^i ( ( cls ` J ) ` A ) ) )
14		eqimss 3561	. . . . . . . 8 |- ( A = ( A i^i ( ( cls ` J ) ` A ) ) -> A C_ ( A i^i ( ( cls ` J ) ` A ) ) )
15	13, 14	syl 16	. . . . . . 7 |- ( ( J e. Top /\ A C_ X ) -> A C_ ( A i^i ( ( cls ` J ) ` A ) ) )
16		sstr 3517	. . . . . . 7 |- ( ( A C_ ( A i^i ( ( cls ` J ) ` A ) ) /\ ( A i^i ( ( cls ` J ) ` A ) ) C_ ( ( int ` J ) ` A ) ) -> A C_ ( ( int ` J ) ` A ) )
17	15, 16	sylan 471	. . . . . 6 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( cls ` J ) ` A ) ) C_ ( ( int ` J ) ` A ) ) -> A C_ ( ( int ` J ) ` A ) )
18	9, 17	sylan2br 476	. . . . 5 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> A C_ ( ( int ` J ) ` A ) )
19	8, 18	eqssd 3526	. . . 4 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> ( ( int ` J ) ` A ) = A )
20	19	ex 434	. . 3 |- ( ( J e. Top /\ A C_ X ) -> ( ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) -> ( ( int ` J ) ` A ) = A ) )
21	5, 20	impbid 191	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( int ` J ) ` A ) = A <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) )
22	6	isopn3 19435	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( A e. J <-> ( ( int ` J ) ` A ) = A ) )
23	6	topbnd 30069	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) = ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) )
24	23	ineq2d 3705	. . 3 |- ( ( J e. Top /\ A C_ X ) -> ( A i^i ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) ) = ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) )
25	24	eqeq1d 2469	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( ( A i^i ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) ) = (/) <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) )
26	21, 22, 25	3bitr4d 285	1 |- ( ( J e. Top /\ A C_ X ) -> ( A e. J <-> ( A i^i ( ( ( cls ` J ) ` A ) i^i ( ( cls ` J ) ` ( X \ A ) ) ) ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   \ cdif 3478   i^i cin 3480   C_ wss 3481   (/)c0 3790   U.cuni 4251   ` cfv 5594   Topctop 19263   intcnt 19386   clsccl 19387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19268  df-cld 19388  df-ntr 19389  df-cls 19390

This theorem is referenced by:  cldbnd  30071