Theorem ressmulr

Description: .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014)

	Ref	Expression
Hypotheses	ressmulr.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
	ressmulr.2	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
Assertion	ressmulr	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_S$$

Proof

Step	Hyp	Ref	Expression
1		ressmulr.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
2		ressmulr.2	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		df-mulr	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}3$$
4		3nn	$$⊢3\in \mathbb{N}$$
5		1lt3	$$⊢1<3$$
6	1 2 3 4 5	resslem	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_S$$