Theorem resshom

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

	Ref	Expression
Hypotheses	resshom.1	$$⊢D=C{\upharpoonright }_{𝑠}A$$
	resshom.2	$$⊢H=\mathrm{Hom}\left(C\right)$$
Assertion	resshom	$$⊢A\in V\to H=\mathrm{Hom}\left(D\right)$$

Proof

Step	Hyp	Ref	Expression
1		resshom.1	$$⊢D=C{\upharpoonright }_{𝑠}A$$
2		resshom.2	$$⊢H=\mathrm{Hom}\left(C\right)$$
3		df-hom	$$⊢\mathrm{Hom}=\mathrm{Slot}14$$
4		1nn0	$$⊢1\in {\mathbb{N}}_0$$
5		4nn	$$⊢4\in \mathbb{N}$$
6	4 5	decnncl	$$⊢14\in \mathbb{N}$$
7		1nn	$$⊢1\in \mathbb{N}$$
8		4nn0	$$⊢4\in {\mathbb{N}}_0$$
9		1lt10	$$⊢1<10$$
10	7 8 4 9	declti	$$⊢1<14$$
11	1 2 3 6 10	resslem	$$⊢A\in V\to H=\mathrm{Hom}\left(D\right)$$