Theorem compssiso 8745

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 4585	. . . . 5 |- ( A e. V -> ( A \ x ) e. _V )
2	1	ralrimivw 2869	. . . 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 5689	. . . 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 8742	. . . . 5 |- `' F = F
7	6	fneq1i 5657	. . . 4 |- ( `' F Fn ~P A <-> F Fn ~P A )
8	5, 7	sylibr 212	. . 3 |- ( A e. V -> `' F Fn ~P A )
9		dff1o4 5806	. . 3 |- ( F : ~P A -1-1-onto-> ~P A <-> ( F Fn ~P A /\ `' F Fn ~P A ) )
10	5, 8, 9	sylanbrc 662	. 2 |- ( A e. V -> F : ~P A -1-1-onto-> ~P A )
11		elpwi 4008	. . . . . . . . 9 |- ( b e. ~P A -> b C_ A )
12	11	ad2antll 726	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> b C_ A )
13	3	isf34lem1 8743	. . . . . . . 8 |- ( ( A e. V /\ b C_ A ) -> ( F ` b ) = ( A \ b ) )
14	12, 13	syldan 468	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` b ) = ( A \ b ) )
15		elpwi 4008	. . . . . . . . 9 |- ( a e. ~P A -> a C_ A )
16	15	ad2antrl 725	. . . . . . . 8 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> a C_ A )
17	3	isf34lem1 8743	. . . . . . . 8 |- ( ( A e. V /\ a C_ A ) -> ( F ` a ) = ( A \ a ) )
18	16, 17	syldan 468	. . . . . . 7 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` a ) = ( A \ a ) )
19	14, 18	psseq12d 3584	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( F ` b ) C. ( F ` a ) <-> ( A \ b ) C. ( A \ a ) ) )
20		difss 3617	. . . . . . 7 |- ( A \ a ) C_ A
21		pssdifcom1 3901	. . . . . . 7 |- ( ( b C_ A /\ ( A \ a ) C_ A ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
22	12, 20, 21	sylancl 660	. . . . . 6 |- ( ( A e. V /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( A \ b ) C. ( A \ a ) <-> ( A \ ( A \ a ) ) C. b ) )
23		dfss4 3729	. . . . . . . 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 3582	. . . . . 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 3109	. . . . . 6 |- b e. _V
28	27	brrpss 6556	. . . . 5 |- ( a [ C.] b <-> a C. b )
29		fvex 5858	. . . . . 6 |- ( F ` a ) e. _V
30	29	brrpss 6556	. . . . 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 6554	. . . . 5 |- Rel [ C.]
33	32	relbrcnv 5365	. . . 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 2875	. 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 5579	. 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 662	1 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   _Vcvv 3106   \ cdif 3458   C_ wss 3461   C. wpss 3462   ~Pcpw 3999   class class class wbr 4439   |-> cmpt 4497   `'ccnv 4987   Fn wfn 5565   -1-1-onto->wf1o 5569   ` cfv 5570   Isom wiso 5571   [ C.] crpss 6552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-rpss 6553

This theorem is referenced by:  isf34lem3  8746  isf34lem5  8749  isfin1-4  8758