Theorem ntrneineine0lem 38381

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-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 )
ntrnei.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
ntrneineine0lem	|- ( ph -> ( E. s e. ~P B X e. ( I ` s ) <-> ( N ` X ) =/= (/) ) )

Distinct variable groups:   B, i, j, k, l, m   k, I, l, m   N, s   X, l, m, s   ph, i, j, k, l, s

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

Proof of Theorem ntrneineine0lem

Step	Hyp	Ref	Expression
1		ntrnei.o	. . . 4 |- 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	. . . 4 |- F = ( ~P B O B )
3		ntrnei.r	. . . . 5 |- ( ph -> I F N )
4	3	adantr 481	. . . 4 |- ( ( ph /\ s e. ~P B ) -> I F N )
5		ntrnei.x	. . . . 5 |- ( ph -> X e. B )
6	5	adantr 481	. . . 4 |- ( ( ph /\ s e. ~P B ) -> X e. B )
7		simpr 477	. . . 4 |- ( ( ph /\ s e. ~P B ) -> s e. ~P B )
8	1, 2, 4, 6, 7	ntrneiel 38379	. . 3 |- ( ( ph /\ s e. ~P B ) -> ( X e. ( I ` s ) <-> s e. ( N ` X ) ) )
9	8	rexbidva 3049	. 2 |- ( ph -> ( E. s e. ~P B X e. ( I ` s ) <-> E. s e. ~P B s e. ( N ` X ) ) )
10	1, 2, 3	ntrneinex 38375	. . . . . . . . . 10 |- ( ph -> N e. ( ~P ~P B ^m B ) )
11		elmapi 7879	. . . . . . . . . 10 |- ( N e. ( ~P ~P B ^m B ) -> N : B --> ~P ~P B )
12	10, 11	syl 17	. . . . . . . . 9 |- ( ph -> N : B --> ~P ~P B )
13	12, 5	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( N ` X ) e. ~P ~P B )
14	13	elpwid 4170	. . . . . . 7 |- ( ph -> ( N ` X ) C_ ~P B )
15	14	sseld 3602	. . . . . 6 |- ( ph -> ( s e. ( N ` X ) -> s e. ~P B ) )
16	15	pm4.71rd 667	. . . . 5 |- ( ph -> ( s e. ( N ` X ) <-> ( s e. ~P B /\ s e. ( N ` X ) ) ) )
17	16	exbidv 1850	. . . 4 |- ( ph -> ( E. s s e. ( N ` X ) <-> E. s ( s e. ~P B /\ s e. ( N ` X ) ) ) )
18	17	bicomd 213	. . 3 |- ( ph -> ( E. s ( s e. ~P B /\ s e. ( N ` X ) ) <-> E. s s e. ( N ` X ) ) )
19		df-rex 2918	. . 3 |- ( E. s e. ~P B s e. ( N ` X ) <-> E. s ( s e. ~P B /\ s e. ( N ` X ) ) )
20		n0 3931	. . 3 |- ( ( N ` X ) =/= (/) <-> E. s s e. ( N ` X ) )
21	18, 19, 20	3bitr4g 303	. 2 |- ( ph -> ( E. s e. ~P B s e. ( N ` X ) <-> ( N ` X ) =/= (/) ) )
22	9, 21	bitrd 268	1 |- ( ph -> ( E. s e. ~P B X e. ( I ` s ) <-> ( N ` X ) =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   (/)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:  ntrneineine0  38385