Theorem ressds

Description: dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015)

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

Proof

Step	Hyp	Ref	Expression
1		ressds.1	$$⊢H=G{\upharpoonright }_{𝑠}A$$
2		ressds.2	$$⊢D=\mathrm{dist}\left(G\right)$$
3		df-ds	$$⊢\mathrm{dist}=\mathrm{Slot}12$$
4		1nn0	$$⊢1\in {\mathbb{N}}_0$$
5		2nn	$$⊢2\in \mathbb{N}$$
6	4 5	decnncl	$$⊢12\in \mathbb{N}$$
7		1nn	$$⊢1\in \mathbb{N}$$
8		2nn0	$$⊢2\in {\mathbb{N}}_0$$
9		1lt10	$$⊢1<10$$
10	7 8 4 9	declti	$$⊢1<12$$
11	1 2 3 6 10	resslem	$$⊢A\in V\to D=\mathrm{dist}\left(H\right)$$