Theorem neicvgf1o 38412

Description: If neighborhood and convergent functions are related by operator H, it is a one-to-one onto relation. (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
neicvgf1o	|- ( ph -> H : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P ~P B ^m B ) )

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 neicvgf1o

Step	Hyp	Ref	Expression
1		neicvg.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		neicvg.d	. . . . . 6 |- D = ( P ` B )
3		neicvg.h	. . . . . 6 |- H = ( F o. ( D o. G ) )
4		neicvg.r	. . . . . 6 |- ( ph -> N H M )
5	2, 3, 4	neicvgbex 38410	. . . . 5 |- ( ph -> B e. _V )
6		pwexg 4850	. . . . 5 |- ( B e. _V -> ~P B e. _V )
7	5, 6	syl 17	. . . 4 |- ( ph -> ~P B e. _V )
8		neicvg.f	. . . 4 |- F = ( ~P B O B )
9	1, 7, 5, 8	fsovf1od 38310	. . 3 |- ( ph -> F : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P ~P B ^m B ) )
10		neicvg.p	. . . . 5 |- P = ( n e. _V |-> ( p e. ( ~P n ^m ~P n ) |-> ( o e. ~P n |-> ( n \ ( p ` ( n \ o ) ) ) ) ) )
11	10, 2, 5	dssmapf1od 38315	. . . 4 |- ( ph -> D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) )
12		neicvg.g	. . . . 5 |- G = ( B O ~P B )
13	1, 5, 7, 12	fsovf1od 38310	. . . 4 |- ( ph -> G : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P B ^m ~P B ) )
14		f1oco 6159	. . . 4 |- ( ( D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) /\ G : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P B ^m ~P B ) ) -> ( D o. G ) : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P B ^m ~P B ) )
15	11, 13, 14	syl2anc 693	. . 3 |- ( ph -> ( D o. G ) : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P B ^m ~P B ) )
16		f1oco 6159	. . 3 |- ( ( F : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P ~P B ^m B ) /\ ( D o. G ) : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P B ^m ~P B ) ) -> ( F o. ( D o. G ) ) : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P ~P B ^m B ) )
17	9, 15, 16	syl2anc 693	. 2 |- ( ph -> ( F o. ( D o. G ) ) : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P ~P B ^m B ) )
18		f1oeq1 6127	. . 3 |- ( H = ( F o. ( D o. G ) ) -> ( H : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P ~P B ^m B ) <-> ( F o. ( D o. G ) ) : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P ~P B ^m B ) ) )
19	3, 18	ax-mp 5	. 2 |- ( H : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P ~P B ^m B ) <-> ( F o. ( D o. G ) ) : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P ~P B ^m B ) )
20	17, 19	sylibr 224	1 |- ( ph -> H : ( ~P ~P B ^m B ) -1-1-onto-> ( ~P ~P B ^m B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   o. ccom 5118   -1-1-onto->wf1o 5887   ` 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)