Theorem termorcl

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

	Ref	Expression
Assertion	termorcl	⊢ ( 𝑇 ∈ ( TermO ‘ 𝐶 ) → 𝐶 ∈ Cat )

Proof

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