Theorem elringchom

Description: A morphism of unital rings is a function. (Contributed by AV, 14-Feb-2020)

	Ref	Expression
Hypotheses	ringcbas.c	$$⊢C=\mathrm{RingCat}\left(U\right)$$
	ringcbas.b	$$⊢B={\mathrm{Base}}_C$$
	ringcbas.u	$$⊢\phi \to U\in V$$
	ringchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	ringchom.x	$$⊢\phi \to X\in B$$
	ringchom.y	$$⊢\phi \to Y\in B$$
Assertion	elringchom	$$⊢\phi \to \left(F\in \left(XHY\right)\to F:{\mathrm{Base}}_X⟶{\mathrm{Base}}_Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		ringcbas.c	$$⊢C=\mathrm{RingCat}\left(U\right)$$
2		ringcbas.b	$$⊢B={\mathrm{Base}}_C$$
3		ringcbas.u	$$⊢\phi \to U\in V$$
4		ringchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
5		ringchom.x	$$⊢\phi \to X\in B$$
6		ringchom.y	$$⊢\phi \to Y\in B$$
7	1 2 3 4 5 6	ringchom	$$⊢\phi \to XHY=X\mathrm{RingHom}Y$$
8	7	eleq2d	$$⊢\phi \to \left(F\in \left(XHY\right)\leftrightarrow F\in \left(X\mathrm{RingHom}Y\right)\right)$$
9		eqid	$$⊢{\mathrm{Base}}_X={\mathrm{Base}}_X$$
10		eqid	$$⊢{\mathrm{Base}}_Y={\mathrm{Base}}_Y$$
11	9 10	rhmf	$$⊢F\in \left(X\mathrm{RingHom}Y\right)\to F:{\mathrm{Base}}_X⟶{\mathrm{Base}}_Y$$
12	8 11	syl6bi	$$⊢\phi \to \left(F\in \left(XHY\right)\to F:{\mathrm{Base}}_X⟶{\mathrm{Base}}_Y\right)$$