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Metamath Proof Explorer

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 )