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 ∩ CRing )
	crhmsubcALTV.j	⊢ 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) )
Assertion	cringcatALTV	⊢ ( 𝑈 ∈ 𝑉 → ( ( RingCatALTV ‘ 𝑈 ) ↾cat 𝐽 ) ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		crhmsubcALTV.c	⊢ 𝐶 = ( 𝑈 ∩ CRing )
2		crhmsubcALTV.j	⊢ 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) )
3		eqid	⊢ ( ( RingCatALTV ‘ 𝑈 ) ↾cat 𝐽 ) = ( ( RingCatALTV ‘ 𝑈 ) ↾cat 𝐽 )
4	1 2	crhmsubcALTV	⊢ ( 𝑈 ∈ 𝑉 → 𝐽 ∈ ( Subcat ‘ ( RingCatALTV ‘ 𝑈 ) ) )
5	3 4	subccat	⊢ ( 𝑈 ∈ 𝑉 → ( ( RingCatALTV ‘ 𝑈 ) ↾cat 𝐽 ) ∈ Cat )