Theorem dssmapntrcls 38426

Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 31189. (Contributed by RP, 21-Apr-2021.)

Hypotheses

Ref	Expression
dssmapclsntr.x	|- X = U. J
dssmapclsntr.k	|- K = ( cls ` J )
dssmapclsntr.i	|- I = ( int ` J )
dssmapclsntr.o	|- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )
dssmapclsntr.d	|- D = ( O ` X )

Assertion

Ref	Expression
dssmapntrcls	|- ( J e. Top -> I = ( D ` K ) )

Distinct variable groups:   J, b, f, s   f, K, s   X, b, f, s

Allowed substitution hints:   D( f, s, b)   I( f, s, b)   K( b)   O( f, s, b)

Proof of Theorem dssmapntrcls

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vpwex 4849	. . . . . . 7 |- ~P t e. _V
2	1	inex2 4800	. . . . . 6 |- ( J i^i ~P t ) e. _V
3	2	uniex 6953	. . . . 5 |- U. ( J i^i ~P t ) e. _V
4	3	rgenw 2924	. . . 4 |- A. t e. ~P X U. ( J i^i ~P t ) e. _V
5		nfcv 2764	. . . . 5 |- F/_ t ~P X
6	5	fnmptf 6016	. . . 4 |- ( A. t e. ~P X U. ( J i^i ~P t ) e. _V -> ( t e. ~P X |-> U. ( J i^i ~P t ) ) Fn ~P X )
7	4, 6	mp1i 13	. . 3 |- ( J e. Top -> ( t e. ~P X |-> U. ( J i^i ~P t ) ) Fn ~P X )
8		dssmapclsntr.i	. . . . 5 |- I = ( int ` J )
9		dssmapclsntr.x	. . . . . 6 |- X = U. J
10	9	ntrfval 20828	. . . . 5 |- ( J e. Top -> ( int ` J ) = ( t e. ~P X |-> U. ( J i^i ~P t ) ) )
11	8, 10	syl5eq 2668	. . . 4 |- ( J e. Top -> I = ( t e. ~P X |-> U. ( J i^i ~P t ) ) )
12	11	fneq1d 5981	. . 3 |- ( J e. Top -> ( I Fn ~P X <-> ( t e. ~P X |-> U. ( J i^i ~P t ) ) Fn ~P X ) )
13	7, 12	mpbird 247	. 2 |- ( J e. Top -> I Fn ~P X )
14		dssmapclsntr.o	. . . . . 6 |- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )
15		dssmapclsntr.d	. . . . . 6 |- D = ( O ` X )
16	9	topopn 20711	. . . . . 6 |- ( J e. Top -> X e. J )
17	14, 15, 16	dssmapf1od 38315	. . . . 5 |- ( J e. Top -> D : ( ~P X ^m ~P X ) -1-1-onto-> ( ~P X ^m ~P X ) )
18		f1of 6137	. . . . 5 |- ( D : ( ~P X ^m ~P X ) -1-1-onto-> ( ~P X ^m ~P X ) -> D : ( ~P X ^m ~P X ) --> ( ~P X ^m ~P X ) )
19	17, 18	syl 17	. . . 4 |- ( J e. Top -> D : ( ~P X ^m ~P X ) --> ( ~P X ^m ~P X ) )
20		dssmapclsntr.k	. . . . 5 |- K = ( cls ` J )
21	9, 20	clselmap 38425	. . . 4 |- ( J e. Top -> K e. ( ~P X ^m ~P X ) )
22	19, 21	ffvelrnd 6360	. . 3 |- ( J e. Top -> ( D ` K ) e. ( ~P X ^m ~P X ) )
23		elmapfn 7880	. . 3 |- ( ( D ` K ) e. ( ~P X ^m ~P X ) -> ( D ` K ) Fn ~P X )
24	22, 23	syl 17	. 2 |- ( J e. Top -> ( D ` K ) Fn ~P X )
25		elpwi 4168	. . . . 5 |- ( t e. ~P X -> t C_ X )
26	9	ntrval2 20855	. . . . 5 |- ( ( J e. Top /\ t C_ X ) -> ( ( int ` J ) ` t ) = ( X \ ( ( cls ` J ) ` ( X \ t ) ) ) )
27	25, 26	sylan2 491	. . . 4 |- ( ( J e. Top /\ t e. ~P X ) -> ( ( int ` J ) ` t ) = ( X \ ( ( cls ` J ) ` ( X \ t ) ) ) )
28	8	fveq1i 6192	. . . 4 |- ( I ` t ) = ( ( int ` J ) ` t )
29	20	fveq1i 6192	. . . . 5 |- ( K ` ( X \ t ) ) = ( ( cls ` J ) ` ( X \ t ) )
30	29	difeq2i 3725	. . . 4 |- ( X \ ( K ` ( X \ t ) ) ) = ( X \ ( ( cls ` J ) ` ( X \ t ) ) )
31	27, 28, 30	3eqtr4g 2681	. . 3 |- ( ( J e. Top /\ t e. ~P X ) -> ( I ` t ) = ( X \ ( K ` ( X \ t ) ) ) )
32	16	adantr 481	. . . 4 |- ( ( J e. Top /\ t e. ~P X ) -> X e. J )
33	21	adantr 481	. . . 4 |- ( ( J e. Top /\ t e. ~P X ) -> K e. ( ~P X ^m ~P X ) )
34		eqid 2622	. . . 4 |- ( D ` K ) = ( D ` K )
35		simpr 477	. . . 4 |- ( ( J e. Top /\ t e. ~P X ) -> t e. ~P X )
36		eqid 2622	. . . 4 |- ( ( D ` K ) ` t ) = ( ( D ` K ) ` t )
37	14, 15, 32, 33, 34, 35, 36	dssmapfv3d 38313	. . 3 |- ( ( J e. Top /\ t e. ~P X ) -> ( ( D ` K ) ` t ) = ( X \ ( K ` ( X \ t ) ) ) )
38	31, 37	eqtr4d 2659	. 2 |- ( ( J e. Top /\ t e. ~P X ) -> ( I ` t ) = ( ( D ` K ) ` t ) )
39	13, 24, 38	eqfnfvd 6314	1 |- ( J e. Top -> I = ( D ` K ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by:  dssmapclsntr  38427