Theorem comfffn

Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	comfffn.o	$$⊢O={\mathrm{comp}}_{𝑓}\left(C\right)$$
	comfffn.b	$$⊢B={\mathrm{Base}}_C$$
Assertion	comfffn	$$⊢OFn\left(\left(B\times B\right)\times B\right)$$

Proof

Step	Hyp	Ref	Expression
1		comfffn.o	$$⊢O={\mathrm{comp}}_{𝑓}\left(C\right)$$
2		comfffn.b	$$⊢B={\mathrm{Base}}_C$$
3		eqid	$$⊢\mathrm{Hom}\left(C\right)=\mathrm{Hom}\left(C\right)$$
4		eqid	$$⊢\mathrm{comp}\left(C\right)=\mathrm{comp}\left(C\right)$$
5	1 2 3 4	comfffval	$$⊢O=\left(x\in \left(B\times B\right),y\in B⟼\left(g\in \left(2^{nd}\left(x\right)\mathrm{Hom}\left(C\right)y\right),f\in \mathrm{Hom}\left(C\right)\left(x\right)⟼g\left(x\mathrm{comp}\left(C\right)y\right)f\right)\right)$$
6		ovex	$$⊢2^{nd}\left(x\right)\mathrm{Hom}\left(C\right)y\in \mathrm{V}$$
7		fvex	$$⊢\mathrm{Hom}\left(C\right)\left(x\right)\in \mathrm{V}$$
8	6 7	mpoex	$$⊢\left(g\in \left(2^{nd}\left(x\right)\mathrm{Hom}\left(C\right)y\right),f\in \mathrm{Hom}\left(C\right)\left(x\right)⟼g\left(x\mathrm{comp}\left(C\right)y\right)f\right)\in \mathrm{V}$$
9	5 8	fnmpoi	$$⊢OFn\left(\left(B\times B\right)\times B\right)$$