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Metamath Proof Explorer

Theorem mpteq2ia

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)

Ref Expression
Hypothesis mpteq2ia.1 ⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
Assertion mpteq2ia ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )

Proof

Step Hyp Ref Expression
1 mpteq2ia.1 ⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
2 1 adantl ⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3 2 mpteq2dva ⊢ ( ⊤ → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )
4 3 mptru ⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )