Theorem ovsng

Description: The operation value of a singleton of a nested ordered pair is the last member. (Contributed by Zhi Wang, 22-Oct-2025)

	Ref	Expression
Assertion	ovsng	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐴 { ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ } 𝐵 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		df-ov	⊢ ( 𝐴 { ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ } 𝐵 ) = ( { ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ } ‘ ⟨ 𝐴 , 𝐵 ⟩ )
2		opex	⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ V
3		fvsng	⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ ∈ V ∧ 𝐶 ∈ 𝑉 ) → ( { ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 )
4	2 3	mpan	⊢ ( 𝐶 ∈ 𝑉 → ( { ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 )
5	1 4	eqtrid	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐴 { ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ } 𝐵 ) = 𝐶 )