Theorem mpteq1

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)

	Ref	Expression
Assertion	mpteq1	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2		eqidd	⊢ ( 𝐴 = 𝐵 → 𝐶 = 𝐶 )
3	1 2	mpteq12dv	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )