Metamath Proof Explorer


Theorem dicdmN

Description: Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dicfn.l = ( le ‘ 𝐾 )
dicfn.a 𝐴 = ( Atoms ‘ 𝐾 )
dicfn.h 𝐻 = ( LHyp ‘ 𝐾 )
dicfn.i 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
Assertion dicdmN ( ( 𝐾𝑉𝑊𝐻 ) → dom 𝐼 = { 𝑝𝐴 ∣ ¬ 𝑝 𝑊 } )

Proof

Step Hyp Ref Expression
1 dicfn.l = ( le ‘ 𝐾 )
2 dicfn.a 𝐴 = ( Atoms ‘ 𝐾 )
3 dicfn.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dicfn.i 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
5 1 2 3 4 dicfnN ( ( 𝐾𝑉𝑊𝐻 ) → 𝐼 Fn { 𝑝𝐴 ∣ ¬ 𝑝 𝑊 } )
6 5 fndmd ( ( 𝐾𝑉𝑊𝐻 ) → dom 𝐼 = { 𝑝𝐴 ∣ ¬ 𝑝 𝑊 } )