# Metamath Proof Explorer

## Theorem funcringcsetcALTV2

Description: The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV2.r ${⊢}{R}=\mathrm{RingCat}\left({U}\right)$
funcringcsetcALTV2.s ${⊢}{S}=\mathrm{SetCat}\left({U}\right)$
funcringcsetcALTV2.b ${⊢}{B}={\mathrm{Base}}_{{R}}$
funcringcsetcALTV2.c ${⊢}{C}={\mathrm{Base}}_{{S}}$
funcringcsetcALTV2.u ${⊢}{\phi }\to {U}\in \mathrm{WUni}$
funcringcsetcALTV2.f ${⊢}{\phi }\to {F}=\left({x}\in {B}⟼{\mathrm{Base}}_{{x}}\right)$
funcringcsetcALTV2.g ${⊢}{\phi }\to {G}=\left({x}\in {B},{y}\in {B}⟼{\mathrm{I}↾}_{\left({x}\mathrm{RingHom}{y}\right)}\right)$
Assertion funcringcsetcALTV2 ${⊢}{\phi }\to {F}\left({R}\mathrm{Func}{S}\right){G}$

### Proof

Step Hyp Ref Expression
1 funcringcsetcALTV2.r ${⊢}{R}=\mathrm{RingCat}\left({U}\right)$
2 funcringcsetcALTV2.s ${⊢}{S}=\mathrm{SetCat}\left({U}\right)$
3 funcringcsetcALTV2.b ${⊢}{B}={\mathrm{Base}}_{{R}}$
4 funcringcsetcALTV2.c ${⊢}{C}={\mathrm{Base}}_{{S}}$
5 funcringcsetcALTV2.u ${⊢}{\phi }\to {U}\in \mathrm{WUni}$
6 funcringcsetcALTV2.f ${⊢}{\phi }\to {F}=\left({x}\in {B}⟼{\mathrm{Base}}_{{x}}\right)$
7 funcringcsetcALTV2.g ${⊢}{\phi }\to {G}=\left({x}\in {B},{y}\in {B}⟼{\mathrm{I}↾}_{\left({x}\mathrm{RingHom}{y}\right)}\right)$
8 eqid ${⊢}\mathrm{Hom}\left({R}\right)=\mathrm{Hom}\left({R}\right)$
9 eqid ${⊢}\mathrm{Hom}\left({S}\right)=\mathrm{Hom}\left({S}\right)$
10 eqid ${⊢}\mathrm{Id}\left({R}\right)=\mathrm{Id}\left({R}\right)$
11 eqid ${⊢}\mathrm{Id}\left({S}\right)=\mathrm{Id}\left({S}\right)$
12 eqid ${⊢}\mathrm{comp}\left({R}\right)=\mathrm{comp}\left({R}\right)$
13 eqid ${⊢}\mathrm{comp}\left({S}\right)=\mathrm{comp}\left({S}\right)$
14 1 ringccat ${⊢}{U}\in \mathrm{WUni}\to {R}\in \mathrm{Cat}$
15 5 14 syl ${⊢}{\phi }\to {R}\in \mathrm{Cat}$
16 2 setccat ${⊢}{U}\in \mathrm{WUni}\to {S}\in \mathrm{Cat}$
17 5 16 syl ${⊢}{\phi }\to {S}\in \mathrm{Cat}$
18 1 2 3 4 5 6 funcringcsetcALTV2lem3 ${⊢}{\phi }\to {F}:{B}⟶{C}$
19 1 2 3 4 5 6 7 funcringcsetcALTV2lem4 ${⊢}{\phi }\to {G}Fn\left({B}×{B}\right)$
20 1 2 3 4 5 6 7 funcringcsetcALTV2lem8 ${⊢}\left({\phi }\wedge \left({a}\in {B}\wedge {b}\in {B}\right)\right)\to \left({a}{G}{b}\right):{a}\mathrm{Hom}\left({R}\right){b}⟶{F}\left({a}\right)\mathrm{Hom}\left({S}\right){F}\left({b}\right)$
21 1 2 3 4 5 6 7 funcringcsetcALTV2lem7 ${⊢}\left({\phi }\wedge {a}\in {B}\right)\to \left({a}{G}{a}\right)\left(\mathrm{Id}\left({R}\right)\left({a}\right)\right)=\mathrm{Id}\left({S}\right)\left({F}\left({a}\right)\right)$
22 1 2 3 4 5 6 7 funcringcsetcALTV2lem9 ${⊢}\left({\phi }\wedge \left({a}\in {B}\wedge {b}\in {B}\wedge {c}\in {B}\right)\wedge \left({h}\in \left({a}\mathrm{Hom}\left({R}\right){b}\right)\wedge {k}\in \left({b}\mathrm{Hom}\left({R}\right){c}\right)\right)\right)\to \left({a}{G}{c}\right)\left({k}\left(⟨{a},{b}⟩\mathrm{comp}\left({R}\right){c}\right){h}\right)=\left({b}{G}{c}\right)\left({k}\right)\left(⟨{F}\left({a}\right),{F}\left({b}\right)⟩\mathrm{comp}\left({S}\right){F}\left({c}\right)\right)\left({a}{G}{b}\right)\left({h}\right)$
23 3 4 8 9 10 11 12 13 15 17 18 19 20 21 22 isfuncd ${⊢}{\phi }\to {F}\left({R}\mathrm{Func}{S}\right){G}$