Theorem compsscnv 8215

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 5241	. 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 3427	. . . . 5 |- ( x = y -> ( A \ x ) = ( A \ y ) )
4	3	cbvmptv 4268	. . . 4 |- ( x e. ~P A |-> ( A \ x ) ) = ( y e. ~P A |-> ( A \ y ) )
5		df-mpt 4236	. . . 4 |- ( y e. ~P A |-> ( A \ y ) ) = { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
6	2, 4, 5	3eqtri 2436	. . 3 |- F = { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
7	6	cnveqi 5014	. 2 |- `' F = `' { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
8		df-mpt 4236	. . 3 |- ( x e. ~P A |-> ( A \ x ) ) = { <. x , y >. | ( x e. ~P A /\ y = ( A \ x ) ) }
9		compsscnvlem 8214	. . . . 5 |- ( ( y e. ~P A /\ x = ( A \ y ) ) -> ( x e. ~P A /\ y = ( A \ x ) ) )
10		compsscnvlem 8214	. . . . 5 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( y e. ~P A /\ x = ( A \ y ) ) )
11	9, 10	impbii 181	. . . 4 |- ( ( y e. ~P A /\ x = ( A \ y ) ) <-> ( x e. ~P A /\ y = ( A \ x ) ) )
12	11	opabbii 4240	. . 3 |- { <. x , y >. | ( y e. ~P A /\ x = ( A \ y ) ) } = { <. x , y >. | ( x e. ~P A /\ y = ( A \ x ) ) }
13	8, 2, 12	3eqtr4i 2442	. 2 |- F = { <. x , y >. | ( y e. ~P A /\ x = ( A \ y ) ) }
14	1, 7, 13	3eqtr4i 2442	1 |- `' F = F

Syntax hints:   /\ wa 359   = wceq 1649   e. wcel 1721   \ cdif 3285   ~Pcpw 3767   {copab 4233   e. cmpt 4234   `'ccnv 4844

This theorem is referenced by:  compssiso  8218  isf34lem3  8219  compss  8220  isf34lem5  8222

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-mpt 4236  df-xp 4851  df-rel 4852  df-cnv 4853