Theorem ressco

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

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

Proof

Step	Hyp	Ref	Expression
1		resshom.1	$$⊢D=C{\upharpoonright }_{𝑠}A$$
2		ressco.2	$$⊢\underset{\dot{}}{·}=\mathrm{comp}\left(C\right)$$
3		df-cco	$$⊢\mathrm{comp}=\mathrm{Slot}15$$
4		1nn0	$$⊢1\in {\mathbb{N}}_0$$
5		5nn	$$⊢5\in \mathbb{N}$$
6	4 5	decnncl	$$⊢15\in \mathbb{N}$$
7		1nn	$$⊢1\in \mathbb{N}$$
8		5nn0	$$⊢5\in {\mathbb{N}}_0$$
9		1lt10	$$⊢1<10$$
10	7 8 4 9	declti	$$⊢1<15$$
11	1 2 3 6 10	resslem	$$⊢A\in V\to \underset{\dot{}}{·}=\mathrm{comp}\left(D\right)$$