Theorem opfusgr

Description: A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020)

	Ref	Expression
Assertion	opfusgr	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ FinUSGraph ↔ ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ∧ 𝑉 ∈ Fin ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ )
2	1	isfusgr	⊢ ( ⟨ 𝑉 , 𝐸 ⟩ ∈ FinUSGraph ↔ ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ∧ ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∈ Fin ) )
3		opvtxfv	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) = 𝑉 )
4	3	eleq1d	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∈ Fin ↔ 𝑉 ∈ Fin ) )
5	4	anbi2d	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ∧ ( Vtx ‘ ⟨ 𝑉 , 𝐸 ⟩ ) ∈ Fin ) ↔ ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ∧ 𝑉 ∈ Fin ) ) )
6	2 5	syl5bb	⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( ⟨ 𝑉 , 𝐸 ⟩ ∈ FinUSGraph ↔ ( ⟨ 𝑉 , 𝐸 ⟩ ∈ USGraph ∧ 𝑉 ∈ Fin ) ) )