Theorem rhmfn

Description: The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020)

	Ref	Expression
Assertion	rhmfn	$$⊢\mathrm{RingHom}Fn\left(\mathrm{Ring}\times \mathrm{Ring}\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfrhm2	$$⊢\mathrm{RingHom}=\left(r\in \mathrm{Ring},s\in \mathrm{Ring}⟼\left(r\mathrm{GrpHom}s\right)\cap \left({\mathrm{mulGrp}}_r\mathrm{MndHom}{\mathrm{mulGrp}}_s\right)\right)$$
2		ovex	$$⊢r\mathrm{GrpHom}s\in \mathrm{V}$$
3	2	inex1	$$⊢\left(r\mathrm{GrpHom}s\right)\cap \left({\mathrm{mulGrp}}_r\mathrm{MndHom}{\mathrm{mulGrp}}_s\right)\in \mathrm{V}$$
4	1 3	fnmpoi	$$⊢\mathrm{RingHom}Fn\left(\mathrm{Ring}\times \mathrm{Ring}\right)$$