Theorem mpteq1d

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016)

	Ref	Expression
Hypothesis	mpteq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	mpteq1d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		mpteq1	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )