Theorem mpteq2ia

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)

	Ref	Expression
Hypothesis	mpteq2ia.1	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
Assertion	mpteq2ia	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		mpteq2ia.1	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
2	1	adantl	⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3	2	mpteq2dva	⊢ ( ⊤ → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )
4	3	mptru	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )