Theorem mpteq1df

Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	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	1 2	alrimi	⊢ ( 𝜑 → ∀ 𝑥 𝐴 = 𝐵 )
4		eqid	⊢ 𝐶 = 𝐶
5	4	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 𝐶 = 𝐶
6		mpteq12f	⊢ ( ( ∀ 𝑥 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝐶 = 𝐶 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )
7	3 5 6	sylancl	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )