Theorem rngccatALTV

Description: The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020) (New usage is discouraged.)

	Ref	Expression
Hypothesis	rngccatALTV.c	⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 )
Assertion	rngccatALTV	⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		rngccatALTV.c	⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 )
2		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
3	1 2	rngccatidALTV	⊢ ( 𝑈 ∈ 𝑉 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( I ↾ ( Base ‘ 𝑥 ) ) ) ) )
4	3	simpld	⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat )