Metamath Proof Explorer

Theorem diaval

Description: The partial isomorphism A for a lattice K . Definition of isomorphism map in Crawley p. 120 line 24. (Contributed by NM, 15-Oct-2013)

Ref Expression
Hypotheses diaval.b 
|- B = ( Base ` K )
diaval.l 
|- .<_ = ( le ` K )
diaval.h 
|- H = ( LHyp ` K )
diaval.t 
|- T = ( ( LTrn ` K ) ` W )
diaval.r 
|- R = ( ( trL ` K ) ` W )
diaval.i 
|- I = ( ( DIsoA ` K ) ` W )
Assertion diaval 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = { f e. T | ( R ` f ) .<_ X } )

Proof

Step Hyp Ref Expression
1 diaval.b 
 |-  B = ( Base ` K )
2 diaval.l 
 |-  .<_ = ( le ` K )
3 diaval.h 
 |-  H = ( LHyp ` K )
4 diaval.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 diaval.r 
 |-  R = ( ( trL ` K ) ` W )
6 diaval.i 
 |-  I = ( ( DIsoA ` K ) ` W )
7 1 2 3 4 5 6 diafval 
 |-  ( ( K e. V /\ W e. H ) -> I = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )
8 7 adantr 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> I = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )
9 8 fveq1d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) ` X ) )
10 simpr 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( X e. B /\ X .<_ W ) )
11 breq1 
 |-  ( y = X -> ( y .<_ W <-> X .<_ W ) )
12 11 elrab 
 |-  ( X e. { y e. B | y .<_ W } <-> ( X e. B /\ X .<_ W ) )
13 10 12 sylibr 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> X e. { y e. B | y .<_ W } )
14 breq2 
 |-  ( x = X -> ( ( R ` f ) .<_ x <-> ( R ` f ) .<_ X ) )
15 14 rabbidv 
 |-  ( x = X -> { f e. T | ( R ` f ) .<_ x } = { f e. T | ( R ` f ) .<_ X } )
16 eqid 
 |-  ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } )
17 4 fvexi 
 |-  T e. _V
18 17 rabex 
 |-  { f e. T | ( R ` f ) .<_ X } e. _V
19 15 16 18 fvmpt 
 |-  ( X e. { y e. B | y .<_ W } -> ( ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) ` X ) = { f e. T | ( R ` f ) .<_ X } )
20 13 19 syl 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) ` X ) = { f e. T | ( R ` f ) .<_ X } )
21 9 20 eqtrd 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = { f e. T | ( R ` f ) .<_ X } )