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	|- A. r e. S r e. Ring
	srhmsubc.c	|- C = ( U i^i S )
	srhmsubc.j	|- J = ( r e. C , s e. C |-> ( r RingHom s ) )
Assertion	sringcat	|- ( U e. V -> ( ( RingCat ` U ) |`cat J ) e. Cat )

Proof

Step	Hyp	Ref	Expression
1		srhmsubc.s	|- A. r e. S r e. Ring
2		srhmsubc.c	|- C = ( U i^i S )
3		srhmsubc.j	|- J = ( r e. C , s e. C |-> ( r RingHom s ) )
4		eqid	|- ( ( RingCat ` U ) |`cat J ) = ( ( RingCat ` U ) |`cat J )
5	1 2 3	srhmsubc	|- ( U e. V -> J e. ( Subcat ` ( RingCat ` U ) ) )
6	4 5	subccat	|- ( U e. V -> ( ( RingCat ` U ) |`cat J ) e. Cat )