Theorem edgstruct

Description: The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020)

	Ref	Expression
Hypothesis	edgstruct.s	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑉 ⟩ , ⟨ ( .ef ‘ ndx ) , 𝐸 ⟩ }
Assertion	edgstruct	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( Edg ‘ 𝐺 ) = ran 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		edgstruct.s	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑉 ⟩ , ⟨ ( .ef ‘ ndx ) , 𝐸 ⟩ }
2		edgval	⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
3	1	struct2griedg	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( iEdg ‘ 𝐺 ) = 𝐸 )
4	3	rneqd	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ran ( iEdg ‘ 𝐺 ) = ran 𝐸 )
5	2 4	eqtrid	⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋 ) → ( Edg ‘ 𝐺 ) = ran 𝐸 )