Theorem dibclN

Description: Closure of partial isomorphism B for a lattice K . (Contributed by NM, 8-Mar-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dibcl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dibcl.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dibclN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		dibcl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dibcl.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
3		eqid	⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
4	1 3 2	dibfna	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) )
5		fnfun	⊢ ( 𝐼 Fn dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) → Fun 𝐼 )
6	4 5	syl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → Fun 𝐼 )
7		fvelrn	⊢ ( ( Fun 𝐼 ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )
8	6 7	sylan	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )