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Metamath Proof Explorer

Theorem fvsn

Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994) (Proof shortened by BJ, 25-Feb-2023)

	Ref	Expression
Hypotheses	fvsn.1	⊢ 𝐴 ∈ V
	fvsn.2	⊢ 𝐵 ∈ V
Assertion	fvsn	⊢ ( { ⟨ 𝐴 , 𝐵 ⟩ } ‘ 𝐴 ) = 𝐵

Proof

Step	Hyp	Ref	Expression
1		fvsn.1	⊢ 𝐴 ∈ V
2		fvsn.2	⊢ 𝐵 ∈ V
3		fvsng	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( { ⟨ 𝐴 , 𝐵 ⟩ } ‘ 𝐴 ) = 𝐵 )
4	1 2 3	mp2an	⊢ ( { ⟨ 𝐴 , 𝐵 ⟩ } ‘ 𝐴 ) = 𝐵