Theorem neicvgnvo 38413

Description: If neighborhood and convergent functions are related by operator H, it is its own converse function. (Contributed by RP, 11-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 )

Assertion

Ref	Expression
neicvgnvo	|- ( ph -> `' H = H )

Distinct variable groups:   B, i, j, k, l, m   B, n, o, p   ph, i, j, k, l   ph, n, o, p

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

Proof of Theorem neicvgnvo

Step	Hyp	Ref	Expression
1		neicvg.h	. . . . 5 |- H = ( F o. ( D o. G ) )
2	1	cnveqi 5297	. . . 4 |- `' H = `' ( F o. ( D o. G ) )
3		cnvco 5308	. . . 4 |- `' ( F o. ( D o. G ) ) = ( `' ( D o. G ) o. `' F )
4		cnvco 5308	. . . . 5 |- `' ( D o. G ) = ( `' G o. `' D )
5	4	coeq1i 5281	. . . 4 |- ( `' ( D o. G ) o. `' F ) = ( ( `' G o. `' D ) o. `' F )
6	2, 3, 5	3eqtri 2648	. . 3 |- `' H = ( ( `' G o. `' D ) o. `' F )
7		neicvg.o	. . . . . 6 |- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
8		neicvg.d	. . . . . . 7 |- D = ( P ` B )
9		neicvg.r	. . . . . . 7 |- ( ph -> N H M )
10	8, 1, 9	neicvgbex 38410	. . . . . 6 |- ( ph -> B e. _V )
11		pwexg 4850	. . . . . . 7 |- ( B e. _V -> ~P B e. _V )
12	10, 11	syl 17	. . . . . 6 |- ( ph -> ~P B e. _V )
13		neicvg.g	. . . . . 6 |- G = ( B O ~P B )
14		neicvg.f	. . . . . 6 |- F = ( ~P B O B )
15	7, 10, 12, 13, 14	fsovcnvd 38308	. . . . 5 |- ( ph -> `' G = F )
16		neicvg.p	. . . . . 6 |- P = ( n e. _V |-> ( p e. ( ~P n ^m ~P n ) |-> ( o e. ~P n |-> ( n \ ( p ` ( n \ o ) ) ) ) ) )
17	16, 8, 10	dssmapnvod 38314	. . . . 5 |- ( ph -> `' D = D )
18	15, 17	coeq12d 5286	. . . 4 |- ( ph -> ( `' G o. `' D ) = ( F o. D ) )
19	7, 12, 10, 14, 13	fsovcnvd 38308	. . . 4 |- ( ph -> `' F = G )
20	18, 19	coeq12d 5286	. . 3 |- ( ph -> ( ( `' G o. `' D ) o. `' F ) = ( ( F o. D ) o. G ) )
21	6, 20	syl5eq 2668	. 2 |- ( ph -> `' H = ( ( F o. D ) o. G ) )
22		coass 5654	. . 3 |- ( ( F o. D ) o. G ) = ( F o. ( D o. G ) )
23	22, 1	eqtr4i 2647	. 2 |- ( ( F o. D ) o. G ) = H
24	21, 23	syl6eq 2672	1 |- ( ph -> `' H = H )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   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:  neicvgnvor  38414