Theorem lautcnv 35376

Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Hypothesis

Ref	Expression
lautcnv.i	|- I = ( LAut ` K )

Assertion

Ref	Expression
lautcnv	|- ( ( K e. V /\ F e. I ) -> `' F e. I )

Proof of Theorem lautcnv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
2		lautcnv.i	. . . 4 |- I = ( LAut ` K )
3	1, 2	laut1o 35371	. . 3 |- ( ( K e. V /\ F e. I ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
4		f1ocnv 6149	. . 3 |- ( F : ( Base ` K ) -1-1-onto-> ( Base ` K ) -> `' F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
5	3, 4	syl 17	. 2 |- ( ( K e. V /\ F e. I ) -> `' F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
6		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
7	1, 6, 2	lautcnvle 35375	. . 3 |- ( ( ( K e. V /\ F e. I ) /\ ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) -> ( x ( le ` K ) y <-> ( `' F ` x ) ( le ` K ) ( `' F ` y ) ) )
8	7	ralrimivva 2971	. 2 |- ( ( K e. V /\ F e. I ) -> A. x e. ( Base ` K ) A. y e. ( Base ` K ) ( x ( le ` K ) y <-> ( `' F ` x ) ( le ` K ) ( `' F ` y ) ) )
9	1, 6, 2	islaut 35369	. . 3 |- ( K e. V -> ( `' F e. I <-> ( `' F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) ( x ( le ` K ) y <-> ( `' F ` x ) ( le ` K ) ( `' F ` y ) ) ) ) )
10	9	adantr 481	. 2 |- ( ( K e. V /\ F e. I ) -> ( `' F e. I <-> ( `' F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) ( x ( le ` K ) y <-> ( `' F ` x ) ( le ` K ) ( `' F ` y ) ) ) ) )
11	5, 8, 10	mpbir2and 957	1 |- ( ( K e. V /\ F e. I ) -> `' F e. I )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888   Basecbs 15857   lecple 15948   LAutclaut 35271

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-res 5126  df-ima 5127  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-laut 35275

This theorem is referenced by:  ldilcnv  35401