Theorem sscid

Description: The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017)

	Ref	Expression
Assertion	sscid	⊢ ( ( 𝐻 Fn ( 𝑆 × 𝑆 ) ∧ 𝑆 ∈ 𝑉 ) → 𝐻 ⊆cat 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		fnresdm	⊢ ( 𝐻 Fn ( 𝑆 × 𝑆 ) → ( 𝐻 ↾ ( 𝑆 × 𝑆 ) ) = 𝐻 )
2	1	adantr	⊢ ( ( 𝐻 Fn ( 𝑆 × 𝑆 ) ∧ 𝑆 ∈ 𝑉 ) → ( 𝐻 ↾ ( 𝑆 × 𝑆 ) ) = 𝐻 )
3		sscres	⊢ ( ( 𝐻 Fn ( 𝑆 × 𝑆 ) ∧ 𝑆 ∈ 𝑉 ) → ( 𝐻 ↾ ( 𝑆 × 𝑆 ) ) ⊆cat 𝐻 )
4	2 3	eqbrtrrd	⊢ ( ( 𝐻 Fn ( 𝑆 × 𝑆 ) ∧ 𝑆 ∈ 𝑉 ) → 𝐻 ⊆cat 𝐻 )