Theorem mpteq2da

Description: Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013) (Revised by Mario Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)

	Ref	Expression
Hypotheses	mpteq2da.1	⊢ Ⅎ 𝑥 𝜑
	mpteq2da.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	mpteq2da	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq2da.1	⊢ Ⅎ 𝑥 𝜑
2		mpteq2da.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3		eqidd	⊢ ( 𝜑 → 𝐴 = 𝐴 )
4	1 3 2	mpteq12da	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )