Theorem resvbas

Description: Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018)

	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		df-base	⊢ Base = Slot 1
4		1nn	⊢ 1 ∈ ℕ
5		1re	⊢ 1 ∈ ℝ
6		1lt5	⊢ 1 < 5
7	5 6	ltneii	⊢ 1 ≠ 5
8	1 2 3 4 7	resvlem	⊢ ( 𝐴 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐻 ) )