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	$$⊢\forall r\in Sr\in \mathrm{Ring}$$
	srhmsubc.c	$$⊢C=U\cap S$$
	srhmsubc.j	$$⊢J=\left(r\in C,s\in C⟼r\mathrm{RingHom}s\right)$$
Assertion	sringcat	$$⊢U\in V\to \mathrm{RingCat}\left(U\right){\upharpoonright }_{\mathrm{cat}}J\in \mathrm{Cat}$$

Proof

Step	Hyp	Ref	Expression
1		srhmsubc.s	$$⊢\forall r\in Sr\in \mathrm{Ring}$$
2		srhmsubc.c	$$⊢C=U\cap S$$
3		srhmsubc.j	$$⊢J=\left(r\in C,s\in C⟼r\mathrm{RingHom}s\right)$$
4		eqid	$$⊢\mathrm{RingCat}\left(U\right){\upharpoonright }_{\mathrm{cat}}J=\mathrm{RingCat}\left(U\right){\upharpoonright }_{\mathrm{cat}}J$$
5	1 2 3	srhmsubc	$$⊢U\in V\to J\in \mathrm{Subcat}\left(\mathrm{RingCat}\left(U\right)\right)$$
6	4 5	subccat	$$⊢U\in V\to \mathrm{RingCat}\left(U\right){\upharpoonright }_{\mathrm{cat}}J\in \mathrm{Cat}$$