Theorem mpteq12dv

Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011) (Revised by Mario Carneiro, 16-Dec-2013) Drop ax-10 while shortening its proof. (Revised by Steven Nguyen and Gino Giotto, 1-Dec-2023)

	Ref	Expression
Hypotheses	mpteq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	mpteq12dv.2	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	mpteq12dv	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐶 )
2		mpteq12dv.2	⊢ ( 𝜑 → 𝐵 = 𝐷 )
3		nfv	⊢ Ⅎ 𝑥 𝜑
4	3 1 2	mpteq12df	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )