Theorem isopn3 20870

Description: A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

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

Assertion

Ref	Expression
isopn3	|- ( ( J e. Top /\ S C_ X ) -> ( S e. J <-> ( ( int ` J ) ` S ) = S ) )

Proof of Theorem isopn3

Step	Hyp	Ref	Expression
1		clscld.1	. . . . 5 |- X = U. J
2	1	ntrval 20840	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
3		inss2 3834	. . . . . . . 8 |- ( J i^i ~P S ) C_ ~P S
4	3	unissi 4461	. . . . . . 7 |- U. ( J i^i ~P S ) C_ U. ~P S
5		unipw 4918	. . . . . . 7 |- U. ~P S = S
6	4, 5	sseqtri 3637	. . . . . 6 |- U. ( J i^i ~P S ) C_ S
7	6	a1i 11	. . . . 5 |- ( S e. J -> U. ( J i^i ~P S ) C_ S )
8		id 22	. . . . . . 7 |- ( S e. J -> S e. J )
9		pwidg 4173	. . . . . . 7 |- ( S e. J -> S e. ~P S )
10	8, 9	elind 3798	. . . . . 6 |- ( S e. J -> S e. ( J i^i ~P S ) )
11		elssuni 4467	. . . . . 6 |- ( S e. ( J i^i ~P S ) -> S C_ U. ( J i^i ~P S ) )
12	10, 11	syl 17	. . . . 5 |- ( S e. J -> S C_ U. ( J i^i ~P S ) )
13	7, 12	eqssd 3620	. . . 4 |- ( S e. J -> U. ( J i^i ~P S ) = S )
14	2, 13	sylan9eq 2676	. . 3 |- ( ( ( J e. Top /\ S C_ X ) /\ S e. J ) -> ( ( int ` J ) ` S ) = S )
15	14	ex 450	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( S e. J -> ( ( int ` J ) ` S ) = S ) )
16	1	ntropn 20853	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) e. J )
17		eleq1 2689	. . 3 |- ( ( ( int ` J ) ` S ) = S -> ( ( ( int ` J ) ` S ) e. J <-> S e. J ) )
18	16, 17	syl5ibcom 235	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( ( int ` J ) ` S ) = S -> S e. J ) )
19	15, 18	impbid 202	1 |- ( ( J e. Top /\ S C_ X ) -> ( S e. J <-> ( ( int ` J ) ` S ) = S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ` cfv 5888   Topctop 20698   intcnt 20821

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-iun 4522  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-ntr 20824

This theorem is referenced by:  ntridm  20872  ntrtop  20874  ntr0  20885  isopn3i  20886  opnnei  20924  cnntr  21079  llycmpkgen2  21353  dvnres  23694  dvcnvre  23782  taylthlem2  24128  ulmdvlem3  24156  abelth  24195  opnbnd  32320  ioontr  39736  cncfuni  40099  fperdvper  40133  dirkercncflem3  40322  dirkercncflem4  40323  fourierdlem58  40381  fourierdlem73  40396