Theorem compsscnv 8289

Description: Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)

Hypothesis

Ref	Expression
compss.a	|- F = ( x e. ~P A |-> ( A \ x ) )

Assertion

Ref	Expression
compsscnv	|- `' F = F

Distinct variable group:   x, A

Allowed substitution hint:   F( x)

Proof of Theorem compsscnv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvopab 5309	. 2 |- `' { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) } = { <. x , y >. | ( y e. ~P A /\ x = ( A \ y ) ) }
2		compss.a	. . . 4 |- F = ( x e. ~P A |-> ( A \ x ) )
3		difeq2 3448	. . . . 5 |- ( x = y -> ( A \ x ) = ( A \ y ) )
4	3	cbvmptv 4331	. . . 4 |- ( x e. ~P A |-> ( A \ x ) ) = ( y e. ~P A |-> ( A \ y ) )
5		df-mpt 4299	. . . 4 |- ( y e. ~P A |-> ( A \ y ) ) = { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
6	2, 4, 5	3eqtri 2467	. . 3 |- F = { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
7	6	cnveqi 5082	. 2 |- `' F = `' { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
8		df-mpt 4299	. . 3 |- ( x e. ~P A |-> ( A \ x ) ) = { <. x , y >. | ( x e. ~P A /\ y = ( A \ x ) ) }
9		compsscnvlem 8288	. . . . 5 |- ( ( y e. ~P A /\ x = ( A \ y ) ) -> ( x e. ~P A /\ y = ( A \ x ) ) )
10		compsscnvlem 8288	. . . . 5 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( y e. ~P A /\ x = ( A \ y ) ) )
11	9, 10	impbii 182	. . . 4 |- ( ( y e. ~P A /\ x = ( A \ y ) ) <-> ( x e. ~P A /\ y = ( A \ x ) ) )
12	11	opabbii 4303	. . 3 |- { <. x , y >. | ( y e. ~P A /\ x = ( A \ y ) ) } = { <. x , y >. | ( x e. ~P A /\ y = ( A \ x ) ) }
13	8, 2, 12	3eqtr4i 2473	. 2 |- F = { <. x , y >. | ( y e. ~P A /\ x = ( A \ y ) ) }
14	1, 7, 13	3eqtr4i 2473	1 |- `' F = F

Syntax hints:   /\ wa 360   = wceq 1654   e. wcel 1728   \ cdif 3306   ~Pcpw 3828   {copab 4296   e. cmpt 4297   `'ccnv 4912

This theorem is referenced by:  compssiso  8292  isf34lem3  8293  compss  8294  isf34lem5  8296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-mpt 4299  df-xp 4919  df-rel 4920  df-cnv 4921