Theorem dssmapfvd 38311

Description: Value of the duality operator for self-mappings of subsets of a base set, B. (Contributed by RP, 19-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
dssmapfvd	|- ( ph -> D = ( f e. ( ~P B ^m ~P B ) |-> ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) ) )

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

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

Proof of Theorem dssmapfvd

Step	Hyp	Ref	Expression
1		dssmapfvd.d	. 2 |- D = ( O ` B )
2		dssmapfvd.o	. . . 4 |- O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) )
3	2	a1i 11	. . 3 |- ( ph -> O = ( b e. _V |-> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) ) )
4		pweq 4161	. . . . . 6 |- ( b = B -> ~P b = ~P B )
5	4, 4	oveq12d 6668	. . . . 5 |- ( b = B -> ( ~P b ^m ~P b ) = ( ~P B ^m ~P B ) )
6		id 22	. . . . . . 7 |- ( b = B -> b = B )
7		difeq1 3721	. . . . . . . 8 |- ( b = B -> ( b \ s ) = ( B \ s ) )
8	7	fveq2d 6195	. . . . . . 7 |- ( b = B -> ( f ` ( b \ s ) ) = ( f ` ( B \ s ) ) )
9	6, 8	difeq12d 3729	. . . . . 6 |- ( b = B -> ( b \ ( f ` ( b \ s ) ) ) = ( B \ ( f ` ( B \ s ) ) ) )
10	4, 9	mpteq12dv 4733	. . . . 5 |- ( b = B -> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) = ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) )
11	5, 10	mpteq12dv 4733	. . . 4 |- ( b = B -> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) = ( f e. ( ~P B ^m ~P B ) |-> ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) ) )
12	11	adantl 482	. . 3 |- ( ( ph /\ b = B ) -> ( f e. ( ~P b ^m ~P b ) |-> ( s e. ~P b |-> ( b \ ( f ` ( b \ s ) ) ) ) ) = ( f e. ( ~P B ^m ~P B ) |-> ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) ) )
13		dssmapfvd.b	. . . 4 |- ( ph -> B e. V )
14	13	elexd 3214	. . 3 |- ( ph -> B e. _V )
15		ovex 6678	. . . 4 |- ( ~P B ^m ~P B ) e. _V
16		mptexg 6484	. . . 4 |- ( ( ~P B ^m ~P B ) e. _V -> ( f e. ( ~P B ^m ~P B ) |-> ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) ) e. _V )
17	15, 16	mp1i 13	. . 3 |- ( ph -> ( f e. ( ~P B ^m ~P B ) |-> ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) ) e. _V )
18	3, 12, 14, 17	fvmptd 6288	. 2 |- ( ph -> ( O ` B ) = ( f e. ( ~P B ^m ~P B ) |-> ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) ) )
19	1, 18	syl5eq 2668	1 |- ( ph -> D = ( f e. ( ~P B ^m ~P B ) |-> ( s e. ~P B |-> ( B \ ( f ` ( B \ s ) ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   |-> 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-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-pr 4906

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

This theorem is referenced by:  dssmapfv2d  38312  dssmapnvod  38314  dssmapf1od  38315