Theorem funcsetcestrc

Description: The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 28-Mar-2020)

	Ref	Expression
Hypotheses	funcsetcestrc.s	$$⊢S=\mathrm{SetCat}\left(U\right)$$
	funcsetcestrc.c	$$⊢C={\mathrm{Base}}_S$$
	funcsetcestrc.f	$$⊢\phi \to F=\left(x\in C⟼\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},x⟩\right\}\right)$$
	funcsetcestrc.u	$$⊢\phi \to U\in \mathrm{WUni}$$
	funcsetcestrc.o	$$⊢\phi \to \mathrm{\omega }\in U$$
	funcsetcestrc.g	$$⊢\phi \to G=\left(x\in C,y\in C⟼{\mathrm{I}\upharpoonright }_{\left(y^x\right)}\right)$$
	funcsetcestrc.e	$$⊢E=\mathrm{ExtStrCat}\left(U\right)$$
Assertion	funcsetcestrc	$$⊢\phi \to F\left(S\mathrm{Func}E\right)G$$

Proof

Step	Hyp	Ref	Expression
1		funcsetcestrc.s	$$⊢S=\mathrm{SetCat}\left(U\right)$$
2		funcsetcestrc.c	$$⊢C={\mathrm{Base}}_S$$
3		funcsetcestrc.f	$$⊢\phi \to F=\left(x\in C⟼\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},x⟩\right\}\right)$$
4		funcsetcestrc.u	$$⊢\phi \to U\in \mathrm{WUni}$$
5		funcsetcestrc.o	$$⊢\phi \to \mathrm{\omega }\in U$$
6		funcsetcestrc.g	$$⊢\phi \to G=\left(x\in C,y\in C⟼{\mathrm{I}\upharpoonright }_{\left(y^x\right)}\right)$$
7		funcsetcestrc.e	$$⊢E=\mathrm{ExtStrCat}\left(U\right)$$
8		eqid	$$⊢{\mathrm{Base}}_E={\mathrm{Base}}_E$$
9		eqid	$$⊢\mathrm{Hom}\left(S\right)=\mathrm{Hom}\left(S\right)$$
10		eqid	$$⊢\mathrm{Hom}\left(E\right)=\mathrm{Hom}\left(E\right)$$
11		eqid	$$⊢\mathrm{Id}\left(S\right)=\mathrm{Id}\left(S\right)$$
12		eqid	$$⊢\mathrm{Id}\left(E\right)=\mathrm{Id}\left(E\right)$$
13		eqid	$$⊢\mathrm{comp}\left(S\right)=\mathrm{comp}\left(S\right)$$
14		eqid	$$⊢\mathrm{comp}\left(E\right)=\mathrm{comp}\left(E\right)$$
15	1	setccat	$$⊢U\in \mathrm{WUni}\to S\in \mathrm{Cat}$$
16	4 15	syl	$$⊢\phi \to S\in \mathrm{Cat}$$
17	7	estrccat	$$⊢U\in \mathrm{WUni}\to E\in \mathrm{Cat}$$
18	4 17	syl	$$⊢\phi \to E\in \mathrm{Cat}$$
19	1 2 3 4 5 7 8	funcsetcestrclem3	$$⊢\phi \to F:C⟶{\mathrm{Base}}_E$$
20	1 2 3 4 5 6	funcsetcestrclem4	$$⊢\phi \to GFn\left(C\times C\right)$$
21	1 2 3 4 5 6 7	funcsetcestrclem8	$$⊢\left(\phi \wedge \left(a\in C\wedge b\in C\right)\right)\to \left(aGb\right):a\mathrm{Hom}\left(S\right)b⟶F\left(a\right)\mathrm{Hom}\left(E\right)F\left(b\right)$$
22	1 2 3 4 5 6 7	funcsetcestrclem7	$$⊢\left(\phi \wedge a\in C\right)\to \left(aGa\right)\left(\mathrm{Id}\left(S\right)\left(a\right)\right)=\mathrm{Id}\left(E\right)\left(F\left(a\right)\right)$$
23	1 2 3 4 5 6 7	funcsetcestrclem9	$$⊢\left(\phi \wedge \left(a\in C\wedge b\in C\wedge c\in C\right)\wedge \left(h\in \left(a\mathrm{Hom}\left(S\right)b\right)\wedge k\in \left(b\mathrm{Hom}\left(S\right)c\right)\right)\right)\to \left(aGc\right)\left(k\left(⟨a,b⟩\mathrm{comp}\left(S\right)c\right)h\right)=\left(bGc\right)\left(k\right)\left(⟨F\left(a\right),F\left(b\right)⟩\mathrm{comp}\left(E\right)F\left(c\right)\right)\left(aGb\right)\left(h\right)$$
24	2 8 9 10 11 12 13 14 16 18 19 20 21 22 23	isfuncd	$$⊢\phi \to F\left(S\mathrm{Func}E\right)G$$