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		mulridx	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		basendxnmulrndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
6	1 2 3 5	resseqnbas	$$⊢A\in V\to \underset{\dot{}}{·}={\cdot }_S$$