Theorem resvbas

Description: Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018) (Revised by AV, 31-Oct-2024)

	Ref	Expression
Hypotheses	resvbas.1	⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
	resvbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	resvbas	⊢ ( 𝐴 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		resvbas.1	⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
2		resvbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
3		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		scandxnbasendx	⊢ ( Scalar ‘ ndx ) ≠ ( Base ‘ ndx )
5	4	necomi	⊢ ( Base ‘ ndx ) ≠ ( Scalar ‘ ndx )
6	1 2 3 5	resvlem	⊢ ( 𝐴 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐻 ) )