This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dilsetN

Description: The set of dilations for a fiducial atom D . (Contributed by NM, 4-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses dilset.a 
|- A = ( Atoms ` K )
dilset.s 
|- S = ( PSubSp ` K )
dilset.w 
|- W = ( WAtoms ` K )
dilset.m 
|- M = ( PAut ` K )
dilset.l 
|- L = ( Dil ` K )
Assertion dilsetN 
|- ( ( K e. B /\ D e. A ) -> ( L ` D ) = { f e. M | A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) } )

Proof

Step Hyp Ref Expression
1 dilset.a 
 |-  A = ( Atoms ` K )
2 dilset.s 
 |-  S = ( PSubSp ` K )
3 dilset.w 
 |-  W = ( WAtoms ` K )
4 dilset.m 
 |-  M = ( PAut ` K )
5 dilset.l 
 |-  L = ( Dil ` K )
6 1 2 3 4 5 dilfsetN 
 |-  ( K e. B -> L = ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) )
7 6 fveq1d 
 |-  ( K e. B -> ( L ` D ) = ( ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) ` D ) )
8 fveq2 
 |-  ( d = D -> ( W ` d ) = ( W ` D ) )
9 8 sseq2d 
 |-  ( d = D -> ( x C_ ( W ` d ) <-> x C_ ( W ` D ) ) )
10 9 imbi1d 
 |-  ( d = D -> ( ( x C_ ( W ` d ) -> ( f ` x ) = x ) <-> ( x C_ ( W ` D ) -> ( f ` x ) = x ) ) )
11 10 ralbidv 
 |-  ( d = D -> ( A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) <-> A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) ) )
12 11 rabbidv 
 |-  ( d = D -> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } = { f e. M | A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) } )
13 eqid 
 |-  ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) = ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } )
14 4 fvexi 
 |-  M e. _V
15 14 rabex 
 |-  { f e. M | A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) } e. _V
16 12 13 15 fvmpt 
 |-  ( D e. A -> ( ( d e. A |-> { f e. M | A. x e. S ( x C_ ( W ` d ) -> ( f ` x ) = x ) } ) ` D ) = { f e. M | A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) } )
17 7 16 sylan9eq 
 |-  ( ( K e. B /\ D e. A ) -> ( L ` D ) = { f e. M | A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) } )