Theorem ressmulr

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

	Ref	Expression
Hypotheses	ressmulr.1	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
	ressmulr.2	⊢ · = ( .r ‘ 𝑅 )
Assertion	ressmulr	⊢ ( 𝐴 ∈ 𝑉 → · = ( .r ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		ressmulr.1	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
2		ressmulr.2	⊢ · = ( .r ‘ 𝑅 )
3		mulridx	⊢ .r = Slot ( .r ‘ ndx )
4		basendxnmulrndx	⊢ ( Base ‘ ndx ) ≠ ( .r ‘ ndx )
5	4	necomi	⊢ ( .r ‘ ndx ) ≠ ( Base ‘ ndx )
6	1 2 3 5	resseqnbas	⊢ ( 𝐴 ∈ 𝑉 → · = ( .r ‘ 𝑆 ) )