Theorem idaf

Description: The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypotheses	idafval.i	$$⊢I={Id}_a\left(C\right)$$
	idafval.b	$$⊢B={\mathrm{Base}}_C$$
	idafval.c	$$⊢\phi \to C\in \mathrm{Cat}$$
	idaf.a	$$⊢A=\mathrm{Arrow}\left(C\right)$$
Assertion	idaf	$$⊢\phi \to I:B⟶A$$

Proof

Step	Hyp	Ref	Expression
1		idafval.i	$$⊢I={Id}_a\left(C\right)$$
2		idafval.b	$$⊢B={\mathrm{Base}}_C$$
3		idafval.c	$$⊢\phi \to C\in \mathrm{Cat}$$
4		idaf.a	$$⊢A=\mathrm{Arrow}\left(C\right)$$
5		otex	$$⊢⟨x,x,\mathrm{Id}\left(C\right)\left(x\right)⟩\in \mathrm{V}$$
6	5	a1i	$$⊢\left(\phi \wedge x\in B\right)\to ⟨x,x,\mathrm{Id}\left(C\right)\left(x\right)⟩\in \mathrm{V}$$
7		eqid	$$⊢\mathrm{Id}\left(C\right)=\mathrm{Id}\left(C\right)$$
8	1 2 3 7	idafval	$$⊢\phi \to I=\left(x\in B⟼⟨x,x,\mathrm{Id}\left(C\right)\left(x\right)⟩\right)$$
9		eqid	$$⊢{Hom}_a\left(C\right)={Hom}_a\left(C\right)$$
10	4 9	homarw	$$⊢x{Hom}_a\left(C\right)x\subseteq A$$
11	3	adantr	$$⊢\left(\phi \wedge x\in B\right)\to C\in \mathrm{Cat}$$
12		simpr	$$⊢\left(\phi \wedge x\in B\right)\to x\in B$$
13	1 2 11 12 9	idahom	$$⊢\left(\phi \wedge x\in B\right)\to I\left(x\right)\in \left(x{Hom}_a\left(C\right)x\right)$$
14	10 13	sselid	$$⊢\left(\phi \wedge x\in B\right)\to I\left(x\right)\in A$$
15	6 8 14	fmpt2d	$$⊢\phi \to I:B⟶A$$