Theorem ntrclsneine0lem 38362

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that at least one (pseudo-)neighborbood of a particular point exists hold equally. (Contributed by RP, 21-May-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 )
ntrclslem0.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
ntrclsneine0lem	|- ( ph -> ( E. s e. ~P B X e. ( I ` s ) <-> E. s e. ~P B -. X e. ( K ` s ) ) )

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

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

Proof of Theorem ntrclsneine0lem

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( s = t -> ( I ` s ) = ( I ` t ) )
2	1	eleq2d 2687	. . 3 |- ( s = t -> ( X e. ( I ` s ) <-> X e. ( I ` t ) ) )
3	2	cbvrexv 3172	. 2 |- ( E. s e. ~P B X e. ( I ` s ) <-> E. t e. ~P B X e. ( I ` t ) )
4		ntrcls.d	. . . . 5 |- D = ( O ` B )
5		ntrcls.r	. . . . 5 |- ( ph -> I D K )
6	4, 5	ntrclsrcomplex 38333	. . . 4 |- ( ph -> ( B \ s ) e. ~P B )
7	6	adantr 481	. . 3 |- ( ( ph /\ s e. ~P B ) -> ( B \ s ) e. ~P B )
8	4, 5	ntrclsrcomplex 38333	. . . . 5 |- ( ph -> ( B \ t ) e. ~P B )
9	8	adantr 481	. . . 4 |- ( ( ph /\ t e. ~P B ) -> ( B \ t ) e. ~P B )
10		difeq2 3722	. . . . . 6 |- ( s = ( B \ t ) -> ( B \ s ) = ( B \ ( B \ t ) ) )
11	10	adantl 482	. . . . 5 |- ( ( ( ph /\ t e. ~P B ) /\ s = ( B \ t ) ) -> ( B \ s ) = ( B \ ( B \ t ) ) )
12		elpwi 4168	. . . . . . 7 |- ( t e. ~P B -> t C_ B )
13		dfss4 3858	. . . . . . 7 |- ( t C_ B <-> ( B \ ( B \ t ) ) = t )
14	12, 13	sylib 208	. . . . . 6 |- ( t e. ~P B -> ( B \ ( B \ t ) ) = t )
15	14	ad2antlr 763	. . . . 5 |- ( ( ( ph /\ t e. ~P B ) /\ s = ( B \ t ) ) -> ( B \ ( B \ t ) ) = t )
16	11, 15	eqtr2d 2657	. . . 4 |- ( ( ( ph /\ t e. ~P B ) /\ s = ( B \ t ) ) -> t = ( B \ s ) )
17	9, 16	rspcedeq2vd 3319	. . 3 |- ( ( ph /\ t e. ~P B ) -> E. s e. ~P B t = ( B \ s ) )
18		fveq2 6191	. . . . . 6 |- ( t = ( B \ s ) -> ( I ` t ) = ( I ` ( B \ s ) ) )
19	18	eleq2d 2687	. . . . 5 |- ( t = ( B \ s ) -> ( X e. ( I ` t ) <-> X e. ( I ` ( B \ s ) ) ) )
20	19	3ad2ant3 1084	. . . 4 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( X e. ( I ` t ) <-> X e. ( I ` ( B \ s ) ) ) )
21		ntrcls.o	. . . . . 6 |- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
22	5	adantr 481	. . . . . 6 |- ( ( ph /\ s e. ~P B ) -> I D K )
23		ntrclslem0.x	. . . . . . 7 |- ( ph -> X e. B )
24	23	adantr 481	. . . . . 6 |- ( ( ph /\ s e. ~P B ) -> X e. B )
25		simpr 477	. . . . . 6 |- ( ( ph /\ s e. ~P B ) -> s e. ~P B )
26	21, 4, 22, 24, 25	ntrclselnel2 38356	. . . . 5 |- ( ( ph /\ s e. ~P B ) -> ( X e. ( I ` ( B \ s ) ) <-> -. X e. ( K ` s ) ) )
27	26	3adant3 1081	. . . 4 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( X e. ( I ` ( B \ s ) ) <-> -. X e. ( K ` s ) ) )
28	20, 27	bitrd 268	. . 3 |- ( ( ph /\ s e. ~P B /\ t = ( B \ s ) ) -> ( X e. ( I ` t ) <-> -. X e. ( K ` s ) ) )
29	7, 17, 28	rexxfrd2 4885	. 2 |- ( ph -> ( E. t e. ~P B X e. ( I ` t ) <-> E. s e. ~P B -. X e. ( K ` s ) ) )
30	3, 29	syl5bb 272	1 |- ( ph -> ( E. s e. ~P B X e. ( I ` s ) <-> E. s e. ~P B -. X e. ( K ` s ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~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:  ntrclsneine0  38363