Theorem compssiso 8822

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 4545	. . . . 5 |- ( A e. V -> ( A \ x ) e. _V )
2	1	ralrimivw 2810	. . . 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 5714	. . . 4 |- ( A. x e. ~P A ( A \ x ) e. _V -> F Fn ~P A )
5	2, 4	syl 17	. . 3 |- ( A e. V -> F Fn ~P A )
6	3	compsscnv 8819	. . . . 5 |- `' F = F
7	6	fneq1i 5680	. . . 4 |- ( `' F Fn ~P A <-> F Fn ~P A )
8	5, 7	sylibr 217	. . 3 |- ( A e. V -> `' F Fn ~P A )
9		dff1o4 5836	. . 3 |- ( F : ~P A -1-1-onto-> ~P A <-> ( F Fn ~P A /\ `' F Fn ~P A ) )
10	5, 8, 9	sylanbrc 677	. 2 |- ( A e. V -> F : ~P A -1-1-onto-> ~P A )
11		elpwi 3951	. . . . . . . . 9 |- ( b e. ~P A -> b C_ A )
12	11	ad2antll 743	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> b C_ A )
13	3	isf34lem1 8820	. . . . . . . 8 |- ( ( A e. V /\ b C_ A ) -> ( F ` b ) = ( A \ b ) )
14	12, 13	syldan 478	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` b ) = ( A \ b ) )
15		elpwi 3951	. . . . . . . . 9 |- ( a e. ~P A -> a C_ A )
16	15	ad2antrl 742	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> a C_ A )
17	3	isf34lem1 8820	. . . . . . . 8 |- ( ( A e. V /\ a C_ A ) -> ( F ` a ) = ( A \ a ) )
18	16, 17	syldan 478	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` a ) = ( A \ a ) )
19	14, 18	psseq12d 3513	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( F ` b ) C. ( F ` a ) <-> ( A \ b ) C. ( A \ a ) ) )
20		difss 3549	. . . . . . 7 |- ( A \ a ) C_ A
21		pssdifcom1 3844	. . . . . . 7 |- ( ( b C_ A /\ ( A \ a ) C_ A ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
22	12, 20, 21	sylancl 675	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
23		dfss4 3668	. . . . . . . 8 |- ( a C_ A <-> ( A \ ( A \ a ) ) = a )
24	16, 23	sylib 201	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( A \ ( A \ a ) ) = a )
25	24	psseq1d 3511	. . . . . 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 288	. . . . 5 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a C. b <-> ( F ` b ) C. ( F ` a ) ) )
27		vex 3034	. . . . . 6 |- b e. _V
28	27	brrpss 6593	. . . . 5 |- ( a [ C.] b <-> a C. b )
29		fvex 5889	. . . . . 6 |- ( F ` a ) e. _V
30	29	brrpss 6593	. . . . 5 |- ( ( F ` b ) [ C.] ( F ` a ) <-> ( F ` b ) C. ( F ` a ) )
31	26, 28, 30	3bitr4g 296	. . . 4 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a [ C.] b <-> ( F ` b ) [ C.] ( F ` a ) ) )
32		relrpss 6591	. . . . 5 |- Rel [ C.]
33	32	relbrcnv 5216	. . . 4 |- ( ( F ` a ) `' [ C.] ( F ` b ) <-> ( F ` b ) [ C.] ( F ` a ) )
34	31, 33	syl6bbr 271	. . 3 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a [ C.] b <-> ( F ` a ) `' [ C.] ( F ` b ) ) )
35	34	ralrimivva 2814	. 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 5598	. 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 677	1 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   \ cdif 3387   C_ wss 3390   C. wpss 3391   ~Pcpw 3942   class class class wbr 4395   |-> cmpt 4454   `'ccnv 4838   Fn wfn 5584   -1-1-onto->wf1o 5588   ` cfv 5589   Isom wiso 5590   [ C.] crpss 6589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-rpss 6590

This theorem is referenced by:  isf34lem3  8823  isf34lem5  8826  isfin1-4  8835