Theorem sringcat

Description: The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020)

	Ref	Expression
Hypotheses	srhmsubc.s	⊢ ∀ 𝑟 ∈ 𝑆 𝑟 ∈ Ring
	srhmsubc.c	⊢ 𝐶 = ( 𝑈 ∩ 𝑆 )
	srhmsubc.j	⊢ 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) )
Assertion	sringcat	⊢ ( 𝑈 ∈ 𝑉 → ( ( RingCat ‘ 𝑈 ) ↾cat 𝐽 ) ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		srhmsubc.s	⊢ ∀ 𝑟 ∈ 𝑆 𝑟 ∈ Ring
2		srhmsubc.c	⊢ 𝐶 = ( 𝑈 ∩ 𝑆 )
3		srhmsubc.j	⊢ 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) )
4		eqid	⊢ ( ( RingCat ‘ 𝑈 ) ↾cat 𝐽 ) = ( ( RingCat ‘ 𝑈 ) ↾cat 𝐽 )
5	1 2 3	srhmsubc	⊢ ( 𝑈 ∈ 𝑉 → 𝐽 ∈ ( Subcat ‘ ( RingCat ‘ 𝑈 ) ) )
6	4 5	subccat	⊢ ( 𝑈 ∈ 𝑉 → ( ( RingCat ‘ 𝑈 ) ↾cat 𝐽 ) ∈ Cat )