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 ( ( 𝐾𝑉𝑊𝐻 ) → dom 𝐼 = { 𝑥𝐵𝑥 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 ( ( 𝐾𝑉𝑊𝐻 ) → 𝐼 Fn { 𝑥𝐵𝑥 𝑊 } )
6 5 fndmd ( ( 𝐾𝑉𝑊𝐻 ) → dom 𝐼 = { 𝑥𝐵𝑥 𝑊 } )