Theorem dibfnN

Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-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	dibfnN	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 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		eqid	⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
6	3 5 4	dibfna	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) )
7	1 2 3 5	diadm	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } )
8	7	fneq2d	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 Fn dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ↔ 𝐼 Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } ) )
9	6 8	mpbid	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn { 𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊 } )