Theorem mpteq12i

Description: An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010) (Revised by Mario Carneiro, 16-Dec-2013)

	Ref	Expression
Hypotheses	mpteq12i.1	⊢ 𝐴 = 𝐶
	mpteq12i.2	⊢ 𝐵 = 𝐷
Assertion	mpteq12i	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		mpteq12i.1	⊢ 𝐴 = 𝐶
2		mpteq12i.2	⊢ 𝐵 = 𝐷
3	1	a1i	⊢ ( ⊤ → 𝐴 = 𝐶 )
4	2	a1i	⊢ ( ⊤ → 𝐵 = 𝐷 )
5	3 4	mpteq12dv	⊢ ( ⊤ → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )
6	5	mptru	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 )