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Metamath Proof Explorer

Theorem initorcl

Description: Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020)

Ref Expression
Assertion initorcl ( 𝐼 ∈ ( InitO ‘ 𝐶 ) → 𝐶 ∈ Cat )

Proof

Step Hyp Ref Expression
1 df-inito ⊢ InitO = ( 𝑐 ∈ Cat ↦ { 𝑎 ∈ ( Base ‘ 𝑐 ) ∣ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ℎ ℎ ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) } )
2 1 mptrcl ⊢ ( 𝐼 ∈ ( InitO ‘ 𝐶 ) → 𝐶 ∈ Cat )