Theorem ressvsca

Description: .s is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014)

	Ref	Expression
Hypotheses	resssca.1	$$⊢H=G{\upharpoonright }_{𝑠}A$$
	ressvsca.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
Assertion	ressvsca	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_H$$

Proof

Step	Hyp	Ref	Expression
1		resssca.1	$$⊢H=G{\upharpoonright }_{𝑠}A$$
2		ressvsca.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
3		vscaid	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		vscandxnbasendx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
5	1 2 3 4	resseqnbas	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_H$$