Theorem mpteq12df

Description: An equality inference for the maps-to notation. Compare mpteq12dv . (Contributed by Scott Fenton, 8-Aug-2013) (Revised by Mario Carneiro, 11-Dec-2016) (Proof shortened by SN, 11-Nov-2024)

	Ref	Expression
Hypotheses	mpteq12df.1	⊢ Ⅎ 𝑥 𝜑
	mpteq12df.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	mpteq12df.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	mpteq12df	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12df.1	⊢ Ⅎ 𝑥 𝜑
2		mpteq12df.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		mpteq12df.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
4	3	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐷 )
5	1 2 4	mpteq12da	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )