Theorem compssiso 9196

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 4808	. . . . 5 |- ( A e. V -> ( A \ x ) e. _V )
2	1	ralrimivw 2967	. . . 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 6020	. . . 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 9193	. . . . 5 |- `' F = F
7	6	fneq1i 5985	. . . 4 |- ( `' F Fn ~P A <-> F Fn ~P A )
8	5, 7	sylibr 224	. . 3 |- ( A e. V -> `' F Fn ~P A )
9		dff1o4 6145	. . 3 |- ( F : ~P A -1-1-onto-> ~P A <-> ( F Fn ~P A /\ `' F Fn ~P A ) )
10	5, 8, 9	sylanbrc 698	. 2 |- ( A e. V -> F : ~P A -1-1-onto-> ~P A )
11		elpwi 4168	. . . . . . . . 9 |- ( b e. ~P A -> b C_ A )
12	11	ad2antll 765	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> b C_ A )
13	3	isf34lem1 9194	. . . . . . . 8 |- ( ( A e. V /\ b C_ A ) -> ( F ` b ) = ( A \ b ) )
14	12, 13	syldan 487	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` b ) = ( A \ b ) )
15		elpwi 4168	. . . . . . . . 9 |- ( a e. ~P A -> a C_ A )
16	15	ad2antrl 764	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> a C_ A )
17	3	isf34lem1 9194	. . . . . . . 8 |- ( ( A e. V /\ a C_ A ) -> ( F ` a ) = ( A \ a ) )
18	16, 17	syldan 487	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` a ) = ( A \ a ) )
19	14, 18	psseq12d 3701	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( F ` b ) C. ( F ` a ) <-> ( A \ b ) C. ( A \ a ) ) )
20		difss 3737	. . . . . . 7 |- ( A \ a ) C_ A
21		pssdifcom1 4054	. . . . . . 7 |- ( ( b C_ A /\ ( A \ a ) C_ A ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
22	12, 20, 21	sylancl 694	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
23		dfss4 3858	. . . . . . . 8 |- ( a C_ A <-> ( A \ ( A \ a ) ) = a )
24	16, 23	sylib 208	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( A \ ( A \ a ) ) = a )
25	24	psseq1d 3699	. . . . . 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 295	. . . . 5 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a C. b <-> ( F ` b ) C. ( F ` a ) ) )
27		vex 3203	. . . . . 6 |- b e. _V
28	27	brrpss 6940	. . . . 5 |- ( a [ C.] b <-> a C. b )
29		fvex 6201	. . . . . 6 |- ( F ` a ) e. _V
30	29	brrpss 6940	. . . . 5 |- ( ( F ` b ) [ C.] ( F ` a ) <-> ( F ` b ) C. ( F ` a ) )
31	26, 28, 30	3bitr4g 303	. . . 4 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a [ C.] b <-> ( F ` b ) [ C.] ( F ` a ) ) )
32		relrpss 6938	. . . . 5 |- Rel [ C.]
33	32	relbrcnv 5506	. . . 4 |- ( ( F ` a ) `' [ C.] ( F ` b ) <-> ( F ` b ) [ C.] ( F ` a ) )
34	31, 33	syl6bbr 278	. . 3 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a [ C.] b <-> ( F ` a ) `' [ C.] ( F ` b ) ) )
35	34	ralrimivva 2971	. 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 5897	. 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 698	1 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   C. wpss 3575   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   [ C.] crpss 6936

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-rpss 6937

This theorem is referenced by:  isf34lem3  9197  isf34lem5  9200  isfin1-4  9209