Theorem dssmap2d 38316

Description: For any base set B the duality operator for self-mappings of subsets of that base set when composed with itself is the restricted identity operator. (Contributed by RP, 21-Apr-2021.)

Hypotheses

Ref	Expression
dssmapfvd.o	|- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )
dssmapfvd.d	|- D = ( O ` B )
dssmapfvd.b	|- ( ph -> B e. V )

Assertion

Ref	Expression
dssmap2d	|- ( ph -> ( D o. D ) = ( _I |` ( ~P B ^m ~P B ) ) )

Distinct variable groups:   B, b, f, s   ph, b, f, s

Allowed substitution hints:   D( f, s, b)   O( f, s, b)   V( f, s, b)

Proof of Theorem dssmap2d

Step	Hyp	Ref	Expression
1		dssmapfvd.o	. . . 4 |- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )
2		dssmapfvd.d	. . . 4 |- D = ( O ` B )
3		dssmapfvd.b	. . . 4 |- ( ph -> B e. V )
4	1, 2, 3	dssmapnvod 38314	. . 3 |- ( ph -> `' D = D )
5	4	coeq1d 5283	. 2 |- ( ph -> ( `' D o. D ) = ( D o. D ) )
6	1, 2, 3	dssmapf1od 38315	. . 3 |- ( ph -> D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) )
7		f1ococnv1 6165	. . 3 |- ( D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) -> ( `' D o. D ) = ( _I |` ( ~P B ^m ~P B ) ) )
8	6, 7	syl 17	. 2 |- ( ph -> ( `' D o. D ) = ( _I |` ( ~P B ^m ~P B ) ) )
9	5, 8	eqtr3d 2658	1 |- ( ph -> ( D o. D ) = ( _I |` ( ~P B ^m ~P B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   -1-1-onto->wf1o 5887   ` 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: (None)