Theorem subcss1

Description: The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	subcss1.1	$$⊢\phi \to J\in \mathrm{Subcat}\left(C\right)$$
	subcss1.2	$$⊢\phi \to JFn\left(S\times S\right)$$
	subcss1.b	$$⊢B={\mathrm{Base}}_C$$
Assertion	subcss1	$$⊢\phi \to S\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		subcss1.1	$$⊢\phi \to J\in \mathrm{Subcat}\left(C\right)$$
2		subcss1.2	$$⊢\phi \to JFn\left(S\times S\right)$$
3		subcss1.b	$$⊢B={\mathrm{Base}}_C$$
4		eqid	$$⊢{\mathrm{Hom}}_{𝑓}\left(C\right)={\mathrm{Hom}}_{𝑓}\left(C\right)$$
5	4 3	homffn	$$⊢{\mathrm{Hom}}_{𝑓}\left(C\right)Fn\left(B\times B\right)$$
6	5	a1i	$$⊢\phi \to {\mathrm{Hom}}_{𝑓}\left(C\right)Fn\left(B\times B\right)$$
7	1 4	subcssc	$$⊢\phi \to J{\subseteq }_{\mathrm{cat}}{\mathrm{Hom}}_{𝑓}\left(C\right)$$
8	2 6 7	ssc1	$$⊢\phi \to S\subseteq B$$