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	⊢ ( 𝜑 → 𝐽 ∈ ( Subcat ‘ 𝐶 ) )
	subcfn.2	⊢ ( 𝜑 → 𝑆 = dom dom 𝐽 )
Assertion	subcfn	⊢ ( 𝜑 → 𝐽 Fn ( 𝑆 × 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		subcixp.1	⊢ ( 𝜑 → 𝐽 ∈ ( Subcat ‘ 𝐶 ) )
2		subcfn.2	⊢ ( 𝜑 → 𝑆 = dom dom 𝐽 )
3		eqid	⊢ ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐶 )
4	1 3	subcssc	⊢ ( 𝜑 → 𝐽 ⊆cat ( Homf ‘ 𝐶 ) )
5	4 2	sscfn1	⊢ ( 𝜑 → 𝐽 Fn ( 𝑆 × 𝑆 ) )