Theorem mpteq1df

Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021) (Proof shortened by SN, 11-Nov-2024)

	Ref	Expression
Hypotheses	mpteq1df.1	⊢ Ⅎ 𝑥 𝜑
	mpteq1df.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	mpteq1df	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq1df.1	⊢ Ⅎ 𝑥 𝜑
2		mpteq1df.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		eqidd	⊢ ( 𝜑 → 𝐶 = 𝐶 )
4	1 2 3	mpteq12df	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )