Theorem diadm

Description: Domain of the partial isomorphism A. (Contributed by NM, 3-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	diadm	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → dom 𝐼 = { 𝑥 ∈ 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	diafn	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } )
6	5	fndmd	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → dom 𝐼 = { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } )