Theorem resshom

Description: Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017)

	Ref	Expression
Hypotheses	resshom.1	⊢ 𝐷 = ( 𝐶 ↾s 𝐴 )
	resshom.2	⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion	resshom	⊢ ( 𝐴 ∈ 𝑉 → 𝐻 = ( Hom ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		resshom.1	⊢ 𝐷 = ( 𝐶 ↾s 𝐴 )
2		resshom.2	⊢ 𝐻 = ( Hom ‘ 𝐶 )
3		df-hom	⊢ Hom = Slot ; 1 4
4		1nn0	⊢ 1 ∈ ℕ0
5		4nn	⊢ 4 ∈ ℕ
6	4 5	decnncl	⊢ ; 1 4 ∈ ℕ
7		1nn	⊢ 1 ∈ ℕ
8		4nn0	⊢ 4 ∈ ℕ0
9		1lt10	⊢ 1 < ; 1 0
10	7 8 4 9	declti	⊢ 1 < ; 1 4
11	1 2 3 6 10	resslem	⊢ ( 𝐴 ∈ 𝑉 → 𝐻 = ( Hom ‘ 𝐷 ) )