Theorem neicvgel2 38418

Description: The complement of a subset being an element of a neighborhood at a point is equivalent to that subset not being a element of the convergent at that point. (Contributed by RP, 12-Jun-2021.)

Hypotheses

Ref	Expression
neicvg.o	|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
neicvg.p	|- P = ( n e. _V |-> ( p e. ( ~P n ^m ~P n ) |-> ( o e. ~P n |-> ( n \ ( p ` ( n \ o ) ) ) ) ) )
neicvg.d	|- D = ( P ` B )
neicvg.f	|- F = ( ~P B O B )
neicvg.g	|- G = ( B O ~P B )
neicvg.h	|- H = ( F o. ( D o. G ) )
neicvg.r	|- ( ph -> N H M )
neicvgel.x	|- ( ph -> X e. B )
neicvgel.s	|- ( ph -> S e. ~P B )

Assertion

Ref	Expression
neicvgel2	|- ( ph -> ( ( B \ S ) e. ( N ` X ) <-> -. S e. ( M ` X ) ) )

Distinct variable groups:   B, i, j, k, l, m   B, n, o, p   D, i, j, k, l, m   D, n, o, p   i, F, j, k, l   n, F, o, p   i, G, j, k, l, m   n, G, o, p   i, M, j, k, l   n, M, o, p   i, N, j, k, l, m   n, N, o, p   S, m   S, o   X, l, m   ph, i, j, k, l   ph, n, o, p

Allowed substitution hints:   ph( m)   P( i, j, k, m, n, o, p, l)   S( i, j, k, n, p, l)   F( m)   H( i, j, k, m, n, o, p, l)   M( m)   O( i, j, k, m, n, o, p, l)   X( i, j, k, n, o, p)

Proof of Theorem neicvgel2

Step	Hyp	Ref	Expression
1		neicvg.o	. . 3 |- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
2		neicvg.p	. . 3 |- P = ( n e. _V |-> ( p e. ( ~P n ^m ~P n ) |-> ( o e. ~P n |-> ( n \ ( p ` ( n \ o ) ) ) ) ) )
3		neicvg.d	. . 3 |- D = ( P ` B )
4		neicvg.f	. . 3 |- F = ( ~P B O B )
5		neicvg.g	. . 3 |- G = ( B O ~P B )
6		neicvg.h	. . 3 |- H = ( F o. ( D o. G ) )
7		neicvg.r	. . 3 |- ( ph -> N H M )
8		neicvgel.x	. . 3 |- ( ph -> X e. B )
9	3, 6, 7	neicvgrcomplex 38411	. . 3 |- ( ph -> ( B \ S ) e. ~P B )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	neicvgel1 38417	. 2 |- ( ph -> ( ( B \ S ) e. ( N ` X ) <-> -. ( B \ ( B \ S ) ) e. ( M ` X ) ) )
11		neicvgel.s	. . . . . 6 |- ( ph -> S e. ~P B )
12	11	elpwid 4170	. . . . 5 |- ( ph -> S C_ B )
13		dfss4 3858	. . . . 5 |- ( S C_ B <-> ( B \ ( B \ S ) ) = S )
14	12, 13	sylib 208	. . . 4 |- ( ph -> ( B \ ( B \ S ) ) = S )
15	14	eleq1d 2686	. . 3 |- ( ph -> ( ( B \ ( B \ S ) ) e. ( M ` X ) <-> S e. ( M ` X ) ) )
16	15	notbid 308	. 2 |- ( ph -> ( -. ( B \ ( B \ S ) ) e. ( M ` X ) <-> -. S e. ( M ` X ) ) )
17	10, 16	bitrd 268	1 |- ( ph -> ( ( B \ S ) e. ( N ` X ) <-> -. S e. ( M ` X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   o. ccom 5118   ` 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)