Theorem rhmsubccat

Description: The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020)

	Ref	Expression
Hypotheses	rngcrescrhm.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	rngcrescrhm.c	⊢ 𝐶 = ( RngCat ‘ 𝑈 )
	rngcrescrhm.r	⊢ ( 𝜑 → 𝑅 = ( Ring ∩ 𝑈 ) )
	rngcrescrhm.h	⊢ 𝐻 = ( RingHom ↾ ( 𝑅 × 𝑅 ) )
Assertion	rhmsubccat	⊢ ( 𝜑 → ( ( RngCat ‘ 𝑈 ) ↾cat 𝐻 ) ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		rngcrescrhm.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
2		rngcrescrhm.c	⊢ 𝐶 = ( RngCat ‘ 𝑈 )
3		rngcrescrhm.r	⊢ ( 𝜑 → 𝑅 = ( Ring ∩ 𝑈 ) )
4		rngcrescrhm.h	⊢ 𝐻 = ( RingHom ↾ ( 𝑅 × 𝑅 ) )
5		eqid	⊢ ( ( RngCat ‘ 𝑈 ) ↾cat 𝐻 ) = ( ( RngCat ‘ 𝑈 ) ↾cat 𝐻 )
6	1 2 3 4	rhmsubc	⊢ ( 𝜑 → 𝐻 ∈ ( Subcat ‘ ( RngCat ‘ 𝑈 ) ) )
7	5 6	subccat	⊢ ( 𝜑 → ( ( RngCat ‘ 𝑈 ) ↾cat 𝐻 ) ∈ Cat )