Theorem compssiso 8754

Description: Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

Hypothesis

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

Assertion

Ref	Expression
compssiso	|- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )

Distinct variable groups:   x, A   x, V

Allowed substitution hint:   F( x)

Proof of Theorem compssiso

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difexg 4595	. . . . 5 |- ( A e. V -> ( A \ x ) e. _V )
2	1	ralrimivw 2879	. . . 4 |- ( A e. V -> A. x e. ~P A ( A \ x ) e. _V )
3		compss.a	. . . . 5 |- F = ( x e. ~P A |-> ( A \ x ) )
4	3	fnmpt 5707	. . . 4 |- ( A. x e. ~P A ( A \ x ) e. _V -> F Fn ~P A )
5	2, 4	syl 16	. . 3 |- ( A e. V -> F Fn ~P A )
6	3	compsscnv 8751	. . . . 5 |- `' F = F
7	6	fneq1i 5675	. . . 4 |- ( `' F Fn ~P A <-> F Fn ~P A )
8	5, 7	sylibr 212	. . 3 |- ( A e. V -> `' F Fn ~P A )
9		dff1o4 5824	. . 3 |- ( F : ~P A -1-1-onto-> ~P A <-> ( F Fn ~P A /\ `' F Fn ~P A ) )
10	5, 8, 9	sylanbrc 664	. 2 |- ( A e. V -> F : ~P A -1-1-onto-> ~P A )
11		elpwi 4019	. . . . . . . . 9 |- ( b e. ~P A -> b C_ A )
12	11	ad2antll 728	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> b C_ A )
13	3	isf34lem1 8752	. . . . . . . 8 |- ( ( A e. V /\ b C_ A ) -> ( F ` b ) = ( A \ b ) )
14	12, 13	syldan 470	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` b ) = ( A \ b ) )
15		elpwi 4019	. . . . . . . . 9 |- ( a e. ~P A -> a C_ A )
16	15	ad2antrl 727	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> a C_ A )
17	3	isf34lem1 8752	. . . . . . . 8 |- ( ( A e. V /\ a C_ A ) -> ( F ` a ) = ( A \ a ) )
18	16, 17	syldan 470	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` a ) = ( A \ a ) )
19	14, 18	psseq12d 3598	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( F ` b ) C. ( F ` a ) <-> ( A \ b ) C. ( A \ a ) ) )
20		difss 3631	. . . . . . 7 |- ( A \ a ) C_ A
21		pssdifcom1 3912	. . . . . . 7 |- ( ( b C_ A /\ ( A \ a ) C_ A ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
22	12, 20, 21	sylancl 662	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
23		dfss4 3732	. . . . . . . 8 |- ( a C_ A <-> ( A \ ( A \ a ) ) = a )
24	16, 23	sylib 196	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( A \ ( A \ a ) ) = a )
25	24	psseq1d 3596	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( A \ ( A \ a ) ) C. b <-> a C. b ) )
26	19, 22, 25	3bitrrd 280	. . . . 5 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a C. b <-> ( F ` b ) C. ( F ` a ) ) )
27		vex 3116	. . . . . 6 |- b e. _V
28	27	brrpss 6567	. . . . 5 |- ( a [ C.] b <-> a C. b )
29		fvex 5876	. . . . . 6 |- ( F ` a ) e. _V
30	29	brrpss 6567	. . . . 5 |- ( ( F ` b ) [ C.] ( F ` a ) <-> ( F ` b ) C. ( F ` a ) )
31	26, 28, 30	3bitr4g 288	. . . 4 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a [ C.] b <-> ( F ` b ) [ C.] ( F ` a ) ) )
32		relrpss 6565	. . . . 5 |- Rel [ C.]
33	32	relbrcnv 5377	. . . 4 |- ( ( F ` a ) `' [ C.] ( F ` b ) <-> ( F ` b ) [ C.] ( F ` a ) )
34	31, 33	syl6bbr 263	. . 3 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a [ C.] b <-> ( F ` a ) `' [ C.] ( F ` b ) ) )
35	34	ralrimivva 2885	. 2 |- ( A e. V -> A. a e. ~P A A. b e. ~P A ( a [ C.] b <-> ( F ` a ) `' [ C.] ( F ` b ) ) )
36		df-isom 5597	. 2 |- ( F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) <-> ( F : ~P A -1-1-onto-> ~P A /\ A. a e. ~P A A. b e. ~P A ( a [ C.] b <-> ( F ` a ) `' [ C.] ( F ` b ) ) ) )
37	10, 35, 36	sylanbrc 664	1 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   _Vcvv 3113   \ cdif 3473   C_ wss 3476   C. wpss 3477   ~Pcpw 4010   class class class wbr 4447   |-> cmpt 4505   `'ccnv 4998   Fn wfn 5583   -1-1-onto->wf1o 5587   ` cfv 5588   Isom wiso 5589   [ C.] crpss 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-rpss 6564

This theorem is referenced by:  isf34lem3  8755  isf34lem5  8758  isfin1-4  8767