Theorem opnregcld 32325

Description: A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)

Hypothesis

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

Assertion

Ref	Expression
opnregcld	|- ( ( J e. Top /\ A C_ X ) -> ( ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A <-> E. o e. J A = ( ( cls ` J ) ` o ) ) )

Distinct variable groups:   A, o   o, J   o, X

Proof of Theorem opnregcld

Step	Hyp	Ref	Expression
1		opnregcld.1	. . . . 5 |- X = U. J
2	1	ntropn 20853	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) e. J )
3		eqcom 2629	. . . . 5 |- ( ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A <-> A = ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) )
4	3	biimpi 206	. . . 4 |- ( ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A -> A = ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) )
5		fveq2 6191	. . . . . 6 |- ( o = ( ( int ` J ) ` A ) -> ( ( cls ` J ) ` o ) = ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) )
6	5	eqeq2d 2632	. . . . 5 |- ( o = ( ( int ` J ) ` A ) -> ( A = ( ( cls ` J ) ` o ) <-> A = ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) ) )
7	6	rspcev 3309	. . . 4 |- ( ( ( ( int ` J ) ` A ) e. J /\ A = ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) ) -> E. o e. J A = ( ( cls ` J ) ` o ) )
8	2, 4, 7	syl2an 494	. . 3 |- ( ( ( J e. Top /\ A C_ X ) /\ ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A ) -> E. o e. J A = ( ( cls ` J ) ` o ) )
9	8	ex 450	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A -> E. o e. J A = ( ( cls ` J ) ` o ) ) )
10		simpl 473	. . . . . . . 8 |- ( ( J e. Top /\ o e. J ) -> J e. Top )
11	1	eltopss 20712	. . . . . . . . 9 |- ( ( J e. Top /\ o e. J ) -> o C_ X )
12	1	clsss3 20863	. . . . . . . . 9 |- ( ( J e. Top /\ o C_ X ) -> ( ( cls ` J ) ` o ) C_ X )
13	11, 12	syldan 487	. . . . . . . 8 |- ( ( J e. Top /\ o e. J ) -> ( ( cls ` J ) ` o ) C_ X )
14	1	ntrss2 20861	. . . . . . . . 9 |- ( ( J e. Top /\ ( ( cls ` J ) ` o ) C_ X ) -> ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) C_ ( ( cls ` J ) ` o ) )
15	13, 14	syldan 487	. . . . . . . 8 |- ( ( J e. Top /\ o e. J ) -> ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) C_ ( ( cls ` J ) ` o ) )
16	1	clsss 20858	. . . . . . . 8 |- ( ( J e. Top /\ ( ( cls ` J ) ` o ) C_ X /\ ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) C_ ( ( cls ` J ) ` o ) ) -> ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) C_ ( ( cls ` J ) ` ( ( cls ` J ) ` o ) ) )
17	10, 13, 15, 16	syl3anc 1326	. . . . . . 7 |- ( ( J e. Top /\ o e. J ) -> ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) C_ ( ( cls ` J ) ` ( ( cls ` J ) ` o ) ) )
18	1	clsidm 20871	. . . . . . . 8 |- ( ( J e. Top /\ o C_ X ) -> ( ( cls ` J ) ` ( ( cls ` J ) ` o ) ) = ( ( cls ` J ) ` o ) )
19	11, 18	syldan 487	. . . . . . 7 |- ( ( J e. Top /\ o e. J ) -> ( ( cls ` J ) ` ( ( cls ` J ) ` o ) ) = ( ( cls ` J ) ` o ) )
20	17, 19	sseqtrd 3641	. . . . . 6 |- ( ( J e. Top /\ o e. J ) -> ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) C_ ( ( cls ` J ) ` o ) )
21	1	ntrss3 20864	. . . . . . . 8 |- ( ( J e. Top /\ ( ( cls ` J ) ` o ) C_ X ) -> ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) C_ X )
22	13, 21	syldan 487	. . . . . . 7 |- ( ( J e. Top /\ o e. J ) -> ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) C_ X )
23		simpr 477	. . . . . . . 8 |- ( ( J e. Top /\ o e. J ) -> o e. J )
24	1	sscls 20860	. . . . . . . . 9 |- ( ( J e. Top /\ o C_ X ) -> o C_ ( ( cls ` J ) ` o ) )
25	11, 24	syldan 487	. . . . . . . 8 |- ( ( J e. Top /\ o e. J ) -> o C_ ( ( cls ` J ) ` o ) )
26	1	ssntr 20862	. . . . . . . 8 |- ( ( ( J e. Top /\ ( ( cls ` J ) ` o ) C_ X ) /\ ( o e. J /\ o C_ ( ( cls ` J ) ` o ) ) ) -> o C_ ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) )
27	10, 13, 23, 25, 26	syl22anc 1327	. . . . . . 7 |- ( ( J e. Top /\ o e. J ) -> o C_ ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) )
28	1	clsss 20858	. . . . . . 7 |- ( ( J e. Top /\ ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) C_ X /\ o C_ ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) -> ( ( cls ` J ) ` o ) C_ ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) )
29	10, 22, 27, 28	syl3anc 1326	. . . . . 6 |- ( ( J e. Top /\ o e. J ) -> ( ( cls ` J ) ` o ) C_ ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) )
30	20, 29	eqssd 3620	. . . . 5 |- ( ( J e. Top /\ o e. J ) -> ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) = ( ( cls ` J ) ` o ) )
31	30	adantlr 751	. . . 4 |- ( ( ( J e. Top /\ A C_ X ) /\ o e. J ) -> ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) = ( ( cls ` J ) ` o ) )
32		fveq2 6191	. . . . . 6 |- ( A = ( ( cls ` J ) ` o ) -> ( ( int ` J ) ` A ) = ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) )
33	32	fveq2d 6195	. . . . 5 |- ( A = ( ( cls ` J ) ` o ) -> ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) )
34		id 22	. . . . 5 |- ( A = ( ( cls ` J ) ` o ) -> A = ( ( cls ` J ) ` o ) )
35	33, 34	eqeq12d 2637	. . . 4 |- ( A = ( ( cls ` J ) ` o ) -> ( ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A <-> ( ( cls ` J ) ` ( ( int ` J ) ` ( ( cls ` J ) ` o ) ) ) = ( ( cls ` J ) ` o ) ) )
36	31, 35	syl5ibrcom 237	. . 3 |- ( ( ( J e. Top /\ A C_ X ) /\ o e. J ) -> ( A = ( ( cls ` J ) ` o ) -> ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A ) )
37	36	rexlimdva 3031	. 2 |- ( ( J e. Top /\ A C_ X ) -> ( E. o e. J A = ( ( cls ` J ) ` o ) -> ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A ) )
38	9, 37	impbid 202	1 |- ( ( J e. Top /\ A C_ X ) -> ( ( ( cls ` J ) ` ( ( int ` J ) ` A ) ) = A <-> E. o e. J A = ( ( cls ` J ) ` o ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   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: (None)