Theorem elrngchom

Description: A morphism of non-unital rings is a function. (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	elrngchom	$$⊢\phi \to \left(F\in \left(XHY\right)\to F:{\mathrm{Base}}_X⟶{\mathrm{Base}}_Y\right)$$

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 5 6	rngchom	$$⊢\phi \to XHY=X\mathrm{RngHom}Y$$
8	7	eleq2d	$$⊢\phi \to \left(F\in \left(XHY\right)\leftrightarrow F\in \left(X\mathrm{RngHom}Y\right)\right)$$
9		eqid	$$⊢{\mathrm{Base}}_X={\mathrm{Base}}_X$$
10		eqid	$$⊢{\mathrm{Base}}_Y={\mathrm{Base}}_Y$$
11	9 10	rnghmf	$$⊢F\in \left(X\mathrm{RngHom}Y\right)\to F:{\mathrm{Base}}_X⟶{\mathrm{Base}}_Y$$
12	8 11	biimtrdi	$$⊢\phi \to \left(F\in \left(XHY\right)\to F:{\mathrm{Base}}_X⟶{\mathrm{Base}}_Y\right)$$