Theorem mpteq1

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013)

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

Proof

Step	Hyp	Ref	Expression
1		eqidd	⊢ ( 𝑥 ∈ 𝐴 → 𝐶 = 𝐶 )
2	1	rgen	⊢ ∀ 𝑥 ∈ 𝐴 𝐶 = 𝐶
3		mpteq12	⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝐶 = 𝐶 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )
4	2 3	mpan2	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )