Theorem ntrclsfveq1

Description: If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021)

	Ref	Expression
Hypotheses	ntrcls.o	|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
	ntrcls.d	|- D = ( O ` B )
	ntrcls.r	|- ( ph -> I D K )
	ntrclsfv.s	|- ( ph -> S e. ~P B )
	ntrclsfv.c	|- ( ph -> C e. ~P B )
Assertion	ntrclsfveq1	|- ( ph -> ( ( I ` S ) = C <-> ( K ` ( B \ S ) ) = ( B \ C ) ) )

Proof

Step	Hyp	Ref	Expression
1		ntrcls.o	|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
2		ntrcls.d	|- D = ( O ` B )
3		ntrcls.r	|- ( ph -> I D K )
4		ntrclsfv.s	|- ( ph -> S e. ~P B )
5		ntrclsfv.c	|- ( ph -> C e. ~P B )
6	5	elpwid	|- ( ph -> C C_ B )
7		dfss4	|- ( C C_ B <-> ( B \ ( B \ C ) ) = C )
8	6 7	sylib	|- ( ph -> ( B \ ( B \ C ) ) = C )
9	8	eqcomd	|- ( ph -> C = ( B \ ( B \ C ) ) )
10	9	eqeq2d	|- ( ph -> ( ( B \ ( K ` ( B \ S ) ) ) = C <-> ( B \ ( K ` ( B \ S ) ) ) = ( B \ ( B \ C ) ) ) )
11	1 2 3 4	ntrclsfv	|- ( ph -> ( I ` S ) = ( B \ ( K ` ( B \ S ) ) ) )
12	11	eqeq1d	|- ( ph -> ( ( I ` S ) = C <-> ( B \ ( K ` ( B \ S ) ) ) = C ) )
13	1 2 3	ntrclskex	|- ( ph -> K e. ( ~P B ^m ~P B ) )
14		elmapi	|- ( K e. ( ~P B ^m ~P B ) -> K : ~P B --> ~P B )
15	13 14	syl	|- ( ph -> K : ~P B --> ~P B )
16	2 3	ntrclsrcomplex	|- ( ph -> ( B \ S ) e. ~P B )
17	15 16	ffvelcdmd	|- ( ph -> ( K ` ( B \ S ) ) e. ~P B )
18	17	elpwid	|- ( ph -> ( K ` ( B \ S ) ) C_ B )
19		difssd	|- ( ph -> ( B \ C ) C_ B )
20		rcompleq	|- ( ( ( K ` ( B \ S ) ) C_ B /\ ( B \ C ) C_ B ) -> ( ( K ` ( B \ S ) ) = ( B \ C ) <-> ( B \ ( K ` ( B \ S ) ) ) = ( B \ ( B \ C ) ) ) )
21	18 19 20	syl2anc	|- ( ph -> ( ( K ` ( B \ S ) ) = ( B \ C ) <-> ( B \ ( K ` ( B \ S ) ) ) = ( B \ ( B \ C ) ) ) )
22	10 12 21	3bitr4d	|- ( ph -> ( ( I ` S ) = C <-> ( K ` ( B \ S ) ) = ( B \ C ) ) )