Theorem funcringcsetcALTV

Description: The "natural forgetful functor" from the category of 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	funcringcsetcALTV.r	$$⊢R=\mathrm{RingCatALTV}\left(U\right)$$
	funcringcsetcALTV.s	$$⊢S=\mathrm{SetCat}\left(U\right)$$
	funcringcsetcALTV.b	$$⊢B={\mathrm{Base}}_R$$
	funcringcsetcALTV.c	$$⊢C={\mathrm{Base}}_S$$
	funcringcsetcALTV.u	$$⊢\phi \to U\in \mathrm{WUni}$$
	funcringcsetcALTV.f	$$⊢\phi \to F=\left(x\in B⟼{\mathrm{Base}}_x\right)$$
	funcringcsetcALTV.g	$$⊢\phi \to G=\left(x\in B,y\in B⟼{\mathrm{I}\upharpoonright }_{\left(x\mathrm{RingHom}y\right)}\right)$$
Assertion	funcringcsetcALTV	$$⊢\phi \to F\left(R\mathrm{Func}S\right)G$$

Proof

Step	Hyp	Ref	Expression
1		funcringcsetcALTV.r	$$⊢R=\mathrm{RingCatALTV}\left(U\right)$$
2		funcringcsetcALTV.s	$$⊢S=\mathrm{SetCat}\left(U\right)$$
3		funcringcsetcALTV.b	$$⊢B={\mathrm{Base}}_R$$
4		funcringcsetcALTV.c	$$⊢C={\mathrm{Base}}_S$$
5		funcringcsetcALTV.u	$$⊢\phi \to U\in \mathrm{WUni}$$
6		funcringcsetcALTV.f	$$⊢\phi \to F=\left(x\in B⟼{\mathrm{Base}}_x\right)$$
7		funcringcsetcALTV.g	$$⊢\phi \to G=\left(x\in B,y\in B⟼{\mathrm{I}\upharpoonright }_{\left(x\mathrm{RingHom}y\right)}\right)$$
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	ringccatALTV	$$⊢U\in \mathrm{WUni}\to R\in \mathrm{Cat}$$
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	funcringcsetclem3ALTV	$$⊢\phi \to F:B⟶C$$
19	1 2 3 4 5 6 7	funcringcsetclem4ALTV	$$⊢\phi \to GFn\left(B\times B\right)$$
20	1 2 3 4 5 6 7	funcringcsetclem8ALTV	$$⊢\left(\phi \wedge \left(a\in B\wedge b\in B\right)\right)\to \left(aGb\right):a\mathrm{Hom}\left(R\right)b⟶F\left(a\right)\mathrm{Hom}\left(S\right)F\left(b\right)$$
21	1 2 3 4 5 6 7	funcringcsetclem7ALTV	$$⊢\left(\phi \wedge a\in B\right)\to \left(aGa\right)\left(\mathrm{Id}\left(R\right)\left(a\right)\right)=\mathrm{Id}\left(S\right)\left(F\left(a\right)\right)$$
22	1 2 3 4 5 6 7	funcringcsetclem9ALTV	$$⊢\left(\phi \wedge \left(a\in B\wedge b\in B\wedge c\in B\right)\wedge \left(h\in \left(a\mathrm{Hom}\left(R\right)b\right)\wedge k\in \left(b\mathrm{Hom}\left(R\right)c\right)\right)\right)\to \left(aGc\right)\left(k\left(⟨a,b⟩\mathrm{comp}\left(R\right)c\right)h\right)=\left(bGc\right)\left(k\right)\left(⟨F\left(a\right),F\left(b\right)⟩\mathrm{comp}\left(S\right)F\left(c\right)\right)\left(aGb\right)\left(h\right)$$
23	3 4 8 9 10 11 12 13 15 17 18 19 20 21 22	isfuncd	$$⊢\phi \to F\left(R\mathrm{Func}S\right)G$$