Theorem ntrclscls00 38364

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)

Hypotheses

Ref	Expression
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 )

Assertion

Ref	Expression
ntrclscls00	|- ( ph -> ( ( I ` B ) = B <-> ( K ` (/) ) = (/) ) )

Distinct variable groups:   B, i, j, k   j, I, k   j, K, k   ph, i, j, k

Allowed substitution hints:   D( i, j, k)   I( i)   K( i)   O( i, j, k)

Proof of Theorem ntrclscls00

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . . . 6 |- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
2		ntrcls.d	. . . . . 6 |- D = ( O ` B )
3		ntrcls.r	. . . . . 6 |- ( ph -> I D K )
4	1, 2, 3	ntrclsfv1 38353	. . . . 5 |- ( ph -> ( D ` I ) = K )
5	4	fveq1d 6193	. . . 4 |- ( ph -> ( ( D ` I ) ` (/) ) = ( K ` (/) ) )
6	2, 3	ntrclsbex 38332	. . . . 5 |- ( ph -> B e. _V )
7	1, 2, 3	ntrclsiex 38351	. . . . 5 |- ( ph -> I e. ( ~P B ^m ~P B ) )
8		eqid 2622	. . . . 5 |- ( D ` I ) = ( D ` I )
9		0elpw 4834	. . . . . 6 |- (/) e. ~P B
10	9	a1i 11	. . . . 5 |- ( ph -> (/) e. ~P B )
11		eqid 2622	. . . . 5 |- ( ( D ` I ) ` (/) ) = ( ( D ` I ) ` (/) )
12	1, 2, 6, 7, 8, 10, 11	dssmapfv3d 38313	. . . 4 |- ( ph -> ( ( D ` I ) ` (/) ) = ( B \ ( I ` ( B \ (/) ) ) ) )
13	5, 12	eqtr3d 2658	. . 3 |- ( ph -> ( K ` (/) ) = ( B \ ( I ` ( B \ (/) ) ) ) )
14		dif0 3950	. . . . . . 7 |- ( B \ (/) ) = B
15	14	fveq2i 6194	. . . . . 6 |- ( I ` ( B \ (/) ) ) = ( I ` B )
16		id 22	. . . . . 6 |- ( ( I ` B ) = B -> ( I ` B ) = B )
17	15, 16	syl5eq 2668	. . . . 5 |- ( ( I ` B ) = B -> ( I ` ( B \ (/) ) ) = B )
18	17	difeq2d 3728	. . . 4 |- ( ( I ` B ) = B -> ( B \ ( I ` ( B \ (/) ) ) ) = ( B \ B ) )
19		difid 3948	. . . 4 |- ( B \ B ) = (/)
20	18, 19	syl6eq 2672	. . 3 |- ( ( I ` B ) = B -> ( B \ ( I ` ( B \ (/) ) ) ) = (/) )
21	13, 20	sylan9eq 2676	. 2 |- ( ( ph /\ ( I ` B ) = B ) -> ( K ` (/) ) = (/) )
22		pwidg 4173	. . . . 5 |- ( B e. _V -> B e. ~P B )
23	6, 22	syl 17	. . . 4 |- ( ph -> B e. ~P B )
24	1, 2, 3, 23	ntrclsfv 38357	. . 3 |- ( ph -> ( I ` B ) = ( B \ ( K ` ( B \ B ) ) ) )
25	19	fveq2i 6194	. . . . . 6 |- ( K ` ( B \ B ) ) = ( K ` (/) )
26		id 22	. . . . . 6 |- ( ( K ` (/) ) = (/) -> ( K ` (/) ) = (/) )
27	25, 26	syl5eq 2668	. . . . 5 |- ( ( K ` (/) ) = (/) -> ( K ` ( B \ B ) ) = (/) )
28	27	difeq2d 3728	. . . 4 |- ( ( K ` (/) ) = (/) -> ( B \ ( K ` ( B \ B ) ) ) = ( B \ (/) ) )
29	28, 14	syl6eq 2672	. . 3 |- ( ( K ` (/) ) = (/) -> ( B \ ( K ` ( B \ B ) ) ) = B )
30	24, 29	sylan9eq 2676	. 2 |- ( ( ph /\ ( K ` (/) ) = (/) ) -> ( I ` B ) = B )
31	21, 30	impbida 877	1 |- ( ph -> ( ( I ` B ) = B <-> ( K ` (/) ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)