Theorem compssiso 8804

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 4551	. . . . 5 |- ( A e. V -> ( A \ x ) e. _V )
2	1	ralrimivw 2803	. . . 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 5704	. . . 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 8801	. . . . 5 |- `' F = F
7	6	fneq1i 5670	. . . 4 |- ( `' F Fn ~P A <-> F Fn ~P A )
8	5, 7	sylibr 216	. . 3 |- ( A e. V -> `' F Fn ~P A )
9		dff1o4 5822	. . 3 |- ( F : ~P A -1-1-onto-> ~P A <-> ( F Fn ~P A /\ `' F Fn ~P A ) )
10	5, 8, 9	sylanbrc 670	. 2 |- ( A e. V -> F : ~P A -1-1-onto-> ~P A )
11		elpwi 3960	. . . . . . . . 9 |- ( b e. ~P A -> b C_ A )
12	11	ad2antll 735	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> b C_ A )
13	3	isf34lem1 8802	. . . . . . . 8 |- ( ( A e. V /\ b C_ A ) -> ( F ` b ) = ( A \ b ) )
14	12, 13	syldan 473	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` b ) = ( A \ b ) )
15		elpwi 3960	. . . . . . . . 9 |- ( a e. ~P A -> a C_ A )
16	15	ad2antrl 734	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> a C_ A )
17	3	isf34lem1 8802	. . . . . . . 8 |- ( ( A e. V /\ a C_ A ) -> ( F ` a ) = ( A \ a ) )
18	16, 17	syldan 473	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` a ) = ( A \ a ) )
19	14, 18	psseq12d 3527	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( F ` b ) C. ( F ` a ) <-> ( A \ b ) C. ( A \ a ) ) )
20		difss 3560	. . . . . . 7 |- ( A \ a ) C_ A
21		pssdifcom1 3853	. . . . . . 7 |- ( ( b C_ A /\ ( A \ a ) C_ A ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
22	12, 20, 21	sylancl 668	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
23		dfss4 3677	. . . . . . . 8 |- ( a C_ A <-> ( A \ ( A \ a ) ) = a )
24	16, 23	sylib 200	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( A \ ( A \ a ) ) = a )
25	24	psseq1d 3525	. . . . . 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 284	. . . . 5 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a C. b <-> ( F ` b ) C. ( F ` a ) ) )
27		vex 3048	. . . . . 6 |- b e. _V
28	27	brrpss 6574	. . . . 5 |- ( a [ C.] b <-> a C. b )
29		fvex 5875	. . . . . 6 |- ( F ` a ) e. _V
30	29	brrpss 6574	. . . . 5 |- ( ( F ` b ) [ C.] ( F ` a ) <-> ( F ` b ) C. ( F ` a ) )
31	26, 28, 30	3bitr4g 292	. . . 4 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a [ C.] b <-> ( F ` b ) [ C.] ( F ` a ) ) )
32		relrpss 6572	. . . . 5 |- Rel [ C.]
33	32	relbrcnv 5210	. . . 4 |- ( ( F ` a ) `' [ C.] ( F ` b ) <-> ( F ` b ) [ C.] ( F ` a ) )
34	31, 33	syl6bbr 267	. . 3 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a [ C.] b <-> ( F ` a ) `' [ C.] ( F ` b ) ) )
35	34	ralrimivva 2809	. 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 5591	. 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 670	1 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   _Vcvv 3045   \ cdif 3401   C_ wss 3404   C. wpss 3405   ~Pcpw 3951   class class class wbr 4402   |-> cmpt 4461   `'ccnv 4833   Fn wfn 5577   -1-1-onto->wf1o 5581   ` cfv 5582   Isom wiso 5583   [ C.] crpss 6570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-rpss 6571

This theorem is referenced by:  isf34lem3  8805  isf34lem5  8808  isfin1-4  8817