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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → dom 𝐼 = { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ) |
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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ) |
6 | 5 | fndmd | ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → dom 𝐼 = { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ) |