Theorem mpteq12da

Description: An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	mpteq12da.1	⊢ Ⅎ 𝑥 𝜑
	mpteq12da.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	mpteq12da.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐷 )
Assertion	mpteq12da	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12da.1	⊢ Ⅎ 𝑥 𝜑
2		mpteq12da.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		mpteq12da.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐷 )
4	1 2	alrimi	⊢ ( 𝜑 → ∀ 𝑥 𝐴 = 𝐶 )
5	1 3	ralrimia	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐷 )
6		mpteq12f	⊢ ( ( ∀ 𝑥 𝐴 = 𝐶 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐷 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )
7	4 5 6	syl2anc	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )