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

Metamath Proof Explorer

Theorem diaeldm

Description: Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-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 diaeldm 
|- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( 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 1 2 3 4 diadm 
 |-  ( ( K e. V /\ W e. H ) -> dom I = { x e. B | x .<_ W } )
6 5 eleq2d 
 |-  ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. { x e. B | x .<_ W } ) )
7 breq1 
 |-  ( x = X -> ( x .<_ W <-> X .<_ W ) )
8 7 elrab 
 |-  ( X e. { x e. B | x .<_ W } <-> ( X e. B /\ X .<_ W ) )
9 6 8 bitrdi 
 |-  ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( X e. B /\ X .<_ W ) ) )