Theorem compsscnv 9193

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 5533	. 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 3722	. . . . 5 |- ( x = y -> ( A \ x ) = ( A \ y ) )
4	3	cbvmptv 4750	. . . 4 |- ( x e. ~P A |-> ( A \ x ) ) = ( y e. ~P A |-> ( A \ y ) )
5		df-mpt 4730	. . . 4 |- ( y e. ~P A |-> ( A \ y ) ) = { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
6	2, 4, 5	3eqtri 2648	. . 3 |- F = { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
7	6	cnveqi 5297	. 2 |- `' F = `' { <. y , x >. | ( y e. ~P A /\ x = ( A \ y ) ) }
8		df-mpt 4730	. . 3 |- ( x e. ~P A |-> ( A \ x ) ) = { <. x , y >. | ( x e. ~P A /\ y = ( A \ x ) ) }
9		compsscnvlem 9192	. . . . 5 |- ( ( y e. ~P A /\ x = ( A \ y ) ) -> ( x e. ~P A /\ y = ( A \ x ) ) )
10		compsscnvlem 9192	. . . . 5 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( y e. ~P A /\ x = ( A \ y ) ) )
11	9, 10	impbii 199	. . . 4 |- ( ( y e. ~P A /\ x = ( A \ y ) ) <-> ( x e. ~P A /\ y = ( A \ x ) ) )
12	11	opabbii 4717	. . 3 |- { <. x , y >. | ( y e. ~P A /\ x = ( A \ y ) ) } = { <. x , y >. | ( x e. ~P A /\ y = ( A \ x ) ) }
13	8, 2, 12	3eqtr4i 2654	. 2 |- F = { <. x , y >. | ( y e. ~P A /\ x = ( A \ y ) ) }
14	1, 7, 13	3eqtr4i 2654	1 |- `' F = F

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   ~Pcpw 4158   {copab 4712   |-> cmpt 4729   `'ccnv 5113

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-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-ral 2917  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  compssiso  9196  isf34lem3  9197  compss  9198  isf34lem5  9200