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Metamath Proof Explorer

Theorem mpteq1d

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016)

Ref Expression
Hypothesis mpteq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion mpteq1d ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 mpteq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 mpteq1 ⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )
3 1 2 syl ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )