Theorem rngchom

Description: Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020)

	Ref	Expression
Hypotheses	rngcbas.c	$$⊢C=\mathrm{RngCat}\left(U\right)$$
	rngcbas.b	$$⊢B={\mathrm{Base}}_C$$
	rngcbas.u	$$⊢\phi \to U\in V$$
	rngchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	rngchom.x	$$⊢\phi \to X\in B$$
	rngchom.y	$$⊢\phi \to Y\in B$$
Assertion	rngchom	$$⊢\phi \to XHY=X\mathrm{RngHom}Y$$

Proof

Step	Hyp	Ref	Expression
1		rngcbas.c	$$⊢C=\mathrm{RngCat}\left(U\right)$$
2		rngcbas.b	$$⊢B={\mathrm{Base}}_C$$
3		rngcbas.u	$$⊢\phi \to U\in V$$
4		rngchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
5		rngchom.x	$$⊢\phi \to X\in B$$
6		rngchom.y	$$⊢\phi \to Y\in B$$
7	1 2 3 4	rngchomfval	$$⊢\phi \to H={\mathrm{RngHom}\upharpoonright }_{\left(B\times B\right)}$$
8	7	oveqd	$$⊢\phi \to XHY=X\left({\mathrm{RngHom}\upharpoonright }_{\left(B\times B\right)}\right)Y$$
9	5 6	ovresd	$$⊢\phi \to X\left({\mathrm{RngHom}\upharpoonright }_{\left(B\times B\right)}\right)Y=X\mathrm{RngHom}Y$$
10	8 9	eqtrd	$$⊢\phi \to XHY=X\mathrm{RngHom}Y$$