Theorem opvtxfv

Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020)

	Ref	Expression
Assertion	opvtxfv	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		opelvvg	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ⟨ 𝑉 , 𝐸 ⟩ ∈ ( V × V ) )
2		opvtxval	⊢ ( ⟨ 𝑉 , 𝐸 ⟩ ∈ ( V × V ) → ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = ( 1st ‘ ⟨ 𝑉 , 𝐸 ⟩ ) )
3	1 2	syl	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = ( 1st ‘ ⟨ 𝑉 , 𝐸 ⟩ ) )
4		op1stg	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( 1st ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = 𝑉 )
5	3 4	eqtrd	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = 𝑉 )