Theorem opnbnd 32320

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 4040	. . . . 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 3807	. . . . 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 2624	. . . 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 235	. . 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 20861	. . . . . 6 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) C_ A )
8	7	adantr 481	. . . . 5 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> ( ( int ` J ) ` A ) C_ A )
9		inssdif0 3947	. . . . . 6 |- ( ( A i^i ( ( cls ` J ) ` A ) ) C_ ( ( int ` J ) ` A ) <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) )
10	6	sscls 20860	. . . . . . . . . 10 |- ( ( J e. Top /\ A C_ X ) -> A C_ ( ( cls ` J ) ` A ) )
11		df-ss 3588	. . . . . . . . . 10 |- ( A C_ ( ( cls ` J ) ` A ) <-> ( A i^i ( ( cls ` J ) ` A ) ) = A )
12	10, 11	sylib 208	. . . . . . . . 9 |- ( ( J e. Top /\ A C_ X ) -> ( A i^i ( ( cls ` J ) ` A ) ) = A )
13	12	eqcomd 2628	. . . . . . . 8 |- ( ( J e. Top /\ A C_ X ) -> A = ( A i^i ( ( cls ` J ) ` A ) ) )
14		eqimss 3657	. . . . . . . 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 3611	. . . . . . 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 488	. . . . . 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 493	. . . . 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 3620	. . . 4 |- ( ( ( J e. Top /\ A C_ X ) /\ ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) -> ( ( int ` J ) ` A ) = A )
20	19	ex 450	. . 3 |- ( ( J e. Top /\ A C_ X ) -> ( ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) -> ( ( int ` J ) ` A ) = A ) )
21	5, 20	impbid 202	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( int ` J ) ` A ) = A <-> ( A i^i ( ( ( cls ` J ) ` A ) \ ( ( int ` J ) ` A ) ) ) = (/) ) )
22	6	isopn3 20870	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( A e. J <-> ( ( int ` J ) ` A ) = A ) )
23	6	topbnd 32319	. . . 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 3814	. . 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 2624	. 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 300	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 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   intcnt 20821   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:  cldbnd  32321