Theorem drhmsubcALTV

Description: According to df-subc , the subcategories ( SubcatC ) of a category C are subsets of the homomorphisms of C (see subcssc and subcss2 ). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020) (New usage is discouraged.)

	Ref	Expression
Hypotheses	drhmsubcALTV.c	⊢ 𝐶 = ( 𝑈 ∩ DivRing )
	drhmsubcALTV.j	⊢ 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) )
Assertion	drhmsubcALTV	⊢ ( 𝑈 ∈ 𝑉 → 𝐽 ∈ ( Subcat ‘ ( RingCatALTV ‘ 𝑈 ) ) )

Proof

Step	Hyp	Ref	Expression
1		drhmsubcALTV.c	⊢ 𝐶 = ( 𝑈 ∩ DivRing )
2		drhmsubcALTV.j	⊢ 𝐽 = ( 𝑟 ∈ 𝐶 , 𝑠 ∈ 𝐶 ↦ ( 𝑟 RingHom 𝑠 ) )
3		drngring	⊢ ( 𝑟 ∈ DivRing → 𝑟 ∈ Ring )
4	3	rgen	⊢ ∀ 𝑟 ∈ DivRing 𝑟 ∈ Ring
5	4 1 2	srhmsubcALTV	⊢ ( 𝑈 ∈ 𝑉 → 𝐽 ∈ ( Subcat ‘ ( RingCatALTV ‘ 𝑈 ) ) )