Theorem struct2grvtx

Description: The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020)

	Ref	Expression
Hypothesis	struct2grvtx.g	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑉 ⟩ , ⟨ ( .ef ‘ ndx ) , 𝐸 ⟩ }
Assertion	struct2grvtx	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( Vtx ‘ 𝐺 ) = 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		struct2grvtx.g	⊢ 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑉 ⟩ , ⟨ ( .ef ‘ ndx ) , 𝐸 ⟩ }
2		edgfndxnn	⊢ ( .ef ‘ ndx ) ∈ ℕ
3		baseltedgf	⊢ ( Base ‘ ndx ) < ( .ef ‘ ndx )
4	2 3 1	structvtxval	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( Vtx ‘ 𝐺 ) = 𝑉 )