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Metamath Proof Explorer
Description: An equality theorem for the maps-to notation. (Contributed by Mario
Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)
|
|
Ref |
Expression |
|
Assertion |
mpteq1 |
⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
id |
⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 ) |
| 2 |
|
eqidd |
⊢ ( 𝐴 = 𝐵 → 𝐶 = 𝐶 ) |
| 3 |
1 2
|
mpteq12dv |
⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ) |