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Metamath Proof Explorer

Theorem diafn

Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013)

Ref Expression
Hypotheses diafn.b 
|- B = ( Base ` K )
diafn.l 
|- .<_ = ( le ` K )
diafn.h 
|- H = ( LHyp ` K )
diafn.i 
|- I = ( ( DIsoA ` K ) ` W )
Assertion diafn 
|- ( ( K e. V /\ W e. H ) -> I Fn { x e. B | x .<_ W } )

Proof

Step Hyp Ref Expression
1 diafn.b 
 |-  B = ( Base ` K )
2 diafn.l 
 |-  .<_ = ( le ` K )
3 diafn.h 
 |-  H = ( LHyp ` K )
4 diafn.i 
 |-  I = ( ( DIsoA ` K ) ` W )
5 fvex 
 |-  ( ( LTrn ` K ) ` W ) e. _V
6 5 rabex 
 |-  { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } e. _V
7 eqid 
 |-  ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } ) = ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } )
8 6 7 fnmpti 
 |-  ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } ) Fn { x e. B | x .<_ W }
9 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
10 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
11 1 2 3 9 10 4 diafval 
 |-  ( ( K e. V /\ W e. H ) -> I = ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } ) )
12 11 fneq1d 
 |-  ( ( K e. V /\ W e. H ) -> ( I Fn { x e. B | x .<_ W } <-> ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } ) Fn { x e. B | x .<_ W } ) )
13 8 12 mpbiri 
 |-  ( ( K e. V /\ W e. H ) -> I Fn { x e. B | x .<_ W } )