Theorem opnbnd 30993

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 3841	. . . . 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 3629	. . . . 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 2455	. . . 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 224	. . 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 20084	. . . . . 6 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) C_ A )
8	7	adantr 467	. . . . 5 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> ( ( int ` J ) ` A ) C_ A )
9		inssdif0 3836	. . . . . 6 |- ( ( A i^i ( ( cls ` J ) ` A ) ) C_ ( ( int ` J ) ` A ) <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) )
10	6	sscls 20083	. . . . . . . . . 10 |- ( ( J e. Top /\ A C_ X ) -> A C_ ( ( cls ` J ) ` A ) )
11		df-ss 3420	. . . . . . . . . 10 |- ( A C_ ( ( cls ` J ) ` A ) <-> ( A i^i ( ( cls ` J ) ` A ) ) = A )
12	10, 11	sylib 200	. . . . . . . . 9 |- ( ( J e. Top /\ A C_ X ) -> ( A i^i ( ( cls ` J ) ` A ) ) = A )
13	12	eqcomd 2459	. . . . . . . 8 |- ( ( J e. Top /\ A C_ X ) -> A = ( A i^i ( ( cls ` J ) ` A ) ) )
14		eqimss 3486	. . . . . . . 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 3442	. . . . . . 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 474	. . . . . 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 479	. . . . 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 3451	. . . 4 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> ( ( int ` J ) ` A ) = A )
20	19	ex 436	. . 3 |- ( ( J e. Top /\ A C_ X ) -> ( ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) -> ( ( int ` J ) ` A ) = A ) )
21	5, 20	impbid 194	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( int ` J ) ` A ) = A <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) )
22	6	isopn3 20094	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( A e. J <-> ( ( int ` J ) ` A ) = A ) )
23	6	topbnd 30992	. . . 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 3636	. . 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 2455	. 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 289	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 188   /\ wa 371   = wceq 1446   e. wcel 1889   \ cdif 3403   i^i cin 3405   C_ wss 3406   (/)c0 3733   U.cuni 4201   ` cfv 5585   Topctop 19929   intcnt 20044   clsccl 20045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-top 19933  df-cld 20046  df-ntr 20047  df-cls 20048

This theorem is referenced by:  cldbnd  30994