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Metamath Proof Explorer

Theorem dibfnN

Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 dibfn.b 
 |-  B = ( Base ` K )
2 dibfn.l 
 |-  .<_ = ( le ` K )
3 dibfn.h 
 |-  H = ( LHyp ` K )
4 dibfn.i 
 |-  I = ( ( DIsoB ` K ) ` W )
5 eqid 
 |-  ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
6 3 5 4 dibfna 
 |-  ( ( K e. V /\ W e. H ) -> I Fn dom ( ( DIsoA ` K ) ` W ) )
7 1 2 3 5 diadm 
 |-  ( ( K e. V /\ W e. H ) -> dom ( ( DIsoA ` K ) ` W ) = { x e. B | x .<_ W } )
8 7 fneq2d 
 |-  ( ( K e. V /\ W e. H ) -> ( I Fn dom ( ( DIsoA ` K ) ` W ) <-> I Fn { x e. B | x .<_ W } ) )
9 6 8 mpbid 
 |-  ( ( K e. V /\ W e. H ) -> I Fn { x e. B | x .<_ W } )