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Metamath Proof Explorer

Theorem mpteq1df

Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021) (Proof shortened by SN, 11-Nov-2024)

Ref Expression
Hypotheses mpteq1df.1 ⊢ Ⅎ 𝑥 𝜑
mpteq1df.2 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion mpteq1df ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 mpteq1df.1 ⊢ Ⅎ 𝑥 𝜑
2 mpteq1df.2 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
3 eqidd ⊢ ( 𝜑 → 𝐶 = 𝐶 )
4 1 2 3 mpteq12df ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )