Theorem subcfn

Description: An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	subcixp.1	$$⊢\phi \to J\in \mathrm{Subcat}\left(C\right)$$
	subcfn.2	$$⊢\phi \to S=\mathrm{dom}\mathrm{dom}J$$
Assertion	subcfn	$$⊢\phi \to JFn\left(S\times S\right)$$

Proof

Step	Hyp	Ref	Expression
1		subcixp.1	$$⊢\phi \to J\in \mathrm{Subcat}\left(C\right)$$
2		subcfn.2	$$⊢\phi \to S=\mathrm{dom}\mathrm{dom}J$$
3		eqid	$$⊢{\mathrm{Hom}}_{𝑓}\left(C\right)={\mathrm{Hom}}_{𝑓}\left(C\right)$$
4	1 3	subcssc	$$⊢\phi \to J{\subseteq }_{\mathrm{cat}}{\mathrm{Hom}}_{𝑓}\left(C\right)$$
5	4 2	sscfn1	$$⊢\phi \to JFn\left(S\times S\right)$$