Theorem ringchomALTV

Description: Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ringcbasALTV.c	$$⊢C=\mathrm{RingCatALTV}\left(U\right)$$
	ringcbasALTV.b	$$⊢B={\mathrm{Base}}_C$$
	ringcbasALTV.u	$$⊢\phi \to U\in V$$
	ringchomfvalALTV.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	ringchomALTV.x	$$⊢\phi \to X\in B$$
	ringchomALTV.y	$$⊢\phi \to Y\in B$$
Assertion	ringchomALTV	$$⊢\phi \to XHY=X\mathrm{RingHom}Y$$

Proof

Step	Hyp	Ref	Expression
1		ringcbasALTV.c	$$⊢C=\mathrm{RingCatALTV}\left(U\right)$$
2		ringcbasALTV.b	$$⊢B={\mathrm{Base}}_C$$
3		ringcbasALTV.u	$$⊢\phi \to U\in V$$
4		ringchomfvalALTV.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
5		ringchomALTV.x	$$⊢\phi \to X\in B$$
6		ringchomALTV.y	$$⊢\phi \to Y\in B$$
7	1 2 3 4	ringchomfvalALTV	$$⊢\phi \to H=\left(x\in B,y\in B⟼x\mathrm{RingHom}y\right)$$
8		oveq12	$$⊢\left(x=X\wedge y=Y\right)\to x\mathrm{RingHom}y=X\mathrm{RingHom}Y$$
9	8	adantl	$$⊢\left(\phi \wedge \left(x=X\wedge y=Y\right)\right)\to x\mathrm{RingHom}y=X\mathrm{RingHom}Y$$
10		ovexd	$$⊢\phi \to X\mathrm{RingHom}Y\in \mathrm{V}$$
11	7 9 5 6 10	ovmpod	$$⊢\phi \to XHY=X\mathrm{RingHom}Y$$