Theorem cringcatALTV

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

	Ref	Expression
Hypotheses	crhmsubcALTV.c	|- C = ( U i^i CRing )
	crhmsubcALTV.j	|- J = ( r e. C , s e. C |-> ( r RingHom s ) )
Assertion	cringcatALTV	|- ( U e. V -> ( ( RingCatALTV ` U ) |`cat J ) e. Cat )

Proof

Step	Hyp	Ref	Expression
1		crhmsubcALTV.c	|- C = ( U i^i CRing )
2		crhmsubcALTV.j	|- J = ( r e. C , s e. C |-> ( r RingHom s ) )
3		eqid	|- ( ( RingCatALTV ` U ) |`cat J ) = ( ( RingCatALTV ` U ) |`cat J )
4	1 2	crhmsubcALTV	|- ( U e. V -> J e. ( Subcat ` ( RingCatALTV ` U ) ) )
5	3 4	subccat	|- ( U e. V -> ( ( RingCatALTV ` U ) |`cat J ) e. Cat )