Metamath Proof Explorer


Theorem dibdmN

Description: Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-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 dibdmN
|- ( ( K e. V /\ W e. H ) -> dom I = { 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 1 2 3 4 dibfnN
 |-  ( ( K e. V /\ W e. H ) -> I Fn { x e. B | x .<_ W } )
6 5 fndmd
 |-  ( ( K e. V /\ W e. H ) -> dom I = { x e. B | x .<_ W } )