Theorem opnbnd 30766

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 3873	. . . . 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 3663	. . . . 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 2431	. . . 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 223	. . 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 20003	. . . . . 6 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) C_ A )
8	7	adantr 466	. . . . 5 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> ( ( int ` J ) ` A ) C_ A )
9		inssdif0 3868	. . . . . 6 |- ( ( A i^i ( ( cls ` J ) ` A ) ) C_ ( ( int ` J ) ` A ) <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) )
10	6	sscls 20002	. . . . . . . . . 10 |- ( ( J e. Top /\ A C_ X ) -> A C_ ( ( cls ` J ) ` A ) )
11		df-ss 3456	. . . . . . . . . 10 |- ( A C_ ( ( cls ` J ) ` A ) <-> ( A i^i ( ( cls ` J ) ` A ) ) = A )
12	10, 11	sylib 199	. . . . . . . . 9 |- ( ( J e. Top /\ A C_ X ) -> ( A i^i ( ( cls ` J ) ` A ) ) = A )
13	12	eqcomd 2437	. . . . . . . 8 |- ( ( J e. Top /\ A C_ X ) -> A = ( A i^i ( ( cls ` J ) ` A ) ) )
14		eqimss 3522	. . . . . . . 8 |- ( A = ( A i^i ( ( cls ` J ) ` A ) ) -> A C_ ( A i^i ( ( cls ` J ) ` A ) ) )
15	13, 14	syl 17	. . . . . . 7 |- ( ( J e. Top /\ A C_ X ) -> A C_ ( A i^i ( ( cls ` J ) ` A ) ) )
16		sstr 3478	. . . . . . 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 473	. . . . . 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 478	. . . . 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 3487	. . . 4 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> ( ( int ` J ) ` A ) = A )
20	19	ex 435	. . 3 |- ( ( J e. Top /\ A C_ X ) -> ( ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) -> ( ( int ` J ) ` A ) = A ) )
21	5, 20	impbid 193	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( int ` J ) ` A ) = A <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) )
22	6	isopn3 20013	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( A e. J <-> ( ( int ` J ) ` A ) = A ) )
23	6	topbnd 30765	. . . 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 3670	. . 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 2431	. 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 288	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 187   /\ wa 370   = wceq 1437   e. wcel 1870   \ cdif 3439   i^i cin 3441   C_ wss 3442   (/)c0 3767   U.cuni 4222   ` cfv 5601   Topctop 19848   intcnt 19963   clsccl 19964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-top 19852  df-cld 19965  df-ntr 19966  df-cls 19967

This theorem is referenced by:  cldbnd  30767