Theorem ntrneicls11 38388

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that the interior of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)

Hypotheses

Ref	Expression
ntrnei.o	|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
ntrnei.f	|- F = ( ~P B O B )
ntrnei.r	|- ( ph -> I F N )

Assertion

Ref	Expression
ntrneicls11	|- ( ph -> ( ( I ` (/) ) = (/) <-> A. x e. B -. (/) e. ( N ` x ) ) )

Distinct variable groups:   B, i, j, k, l, m, x   k, I, l, m, x   ph, i, j, k, l, x

Allowed substitution hints:   ph( m)   F( x, i, j, k, m, l)   I( i, j)   N( x, i, j, k, m, l)   O( x, i, j, k, m, l)

Proof of Theorem ntrneicls11

Step	Hyp	Ref	Expression
1		ntrnei.o	. . . . . . . . 9 |- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
2		ntrnei.f	. . . . . . . . 9 |- F = ( ~P B O B )
3		ntrnei.r	. . . . . . . . 9 |- ( ph -> I F N )
4	1, 2, 3	ntrneiiex 38374	. . . . . . . 8 |- ( ph -> I e. ( ~P B ^m ~P B ) )
5		elmapi 7879	. . . . . . . 8 |- ( I e. ( ~P B ^m ~P B ) -> I : ~P B --> ~P B )
6	4, 5	syl 17	. . . . . . 7 |- ( ph -> I : ~P B --> ~P B )
7		0elpw 4834	. . . . . . . 8 |- (/) e. ~P B
8	7	a1i 11	. . . . . . 7 |- ( ph -> (/) e. ~P B )
9	6, 8	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( I ` (/) ) e. ~P B )
10	9	elpwid 4170	. . . . 5 |- ( ph -> ( I ` (/) ) C_ B )
11		reldisj 4020	. . . . 5 |- ( ( I ` (/) ) C_ B -> ( ( ( I ` (/) ) i^i B ) = (/) <-> ( I ` (/) ) C_ ( B \ B ) ) )
12	10, 11	syl 17	. . . 4 |- ( ph -> ( ( ( I ` (/) ) i^i B ) = (/) <-> ( I ` (/) ) C_ ( B \ B ) ) )
13	12	bicomd 213	. . 3 |- ( ph -> ( ( I ` (/) ) C_ ( B \ B ) <-> ( ( I ` (/) ) i^i B ) = (/) ) )
14		difid 3948	. . . . 5 |- ( B \ B ) = (/)
15	14	sseq2i 3630	. . . 4 |- ( ( I ` (/) ) C_ ( B \ B ) <-> ( I ` (/) ) C_ (/) )
16		ss0b 3973	. . . 4 |- ( ( I ` (/) ) C_ (/) <-> ( I ` (/) ) = (/) )
17	15, 16	bitri 264	. . 3 |- ( ( I ` (/) ) C_ ( B \ B ) <-> ( I ` (/) ) = (/) )
18		disjr 4018	. . 3 |- ( ( ( I ` (/) ) i^i B ) = (/) <-> A. x e. B -. x e. ( I ` (/) ) )
19	13, 17, 18	3bitr3g 302	. 2 |- ( ph -> ( ( I ` (/) ) = (/) <-> A. x e. B -. x e. ( I ` (/) ) ) )
20	3	adantr 481	. . . . 5 |- ( ( ph /\ x e. B ) -> I F N )
21		simpr 477	. . . . 5 |- ( ( ph /\ x e. B ) -> x e. B )
22	7	a1i 11	. . . . 5 |- ( ( ph /\ x e. B ) -> (/) e. ~P B )
23	1, 2, 20, 21, 22	ntrneiel 38379	. . . 4 |- ( ( ph /\ x e. B ) -> ( x e. ( I ` (/) ) <-> (/) e. ( N ` x ) ) )
24	23	notbid 308	. . 3 |- ( ( ph /\ x e. B ) -> ( -. x e. ( I ` (/) ) <-> -. (/) e. ( N ` x ) ) )
25	24	ralbidva 2985	. 2 |- ( ph -> ( A. x e. B -. x e. ( I ` (/) ) <-> A. x e. B -. (/) e. ( N ` x ) ) )
26	19, 25	bitrd 268	1 |- ( ph -> ( ( I ` (/) ) = (/) <-> A. x e. B -. (/) e. ( N ` x ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^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)