Theorem ressvsca

Description: .s is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014)

	Ref	Expression
Hypotheses	resssca.1	⊢ 𝐻 = ( 𝐺 ↾s 𝐴 )
	ressvsca.2	⊢ · = ( ·𝑠 ‘ 𝐺 )
Assertion	ressvsca	⊢ ( 𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		resssca.1	⊢ 𝐻 = ( 𝐺 ↾s 𝐴 )
2		ressvsca.2	⊢ · = ( ·𝑠 ‘ 𝐺 )
3		vscaid	⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx )
4		vscandxnbasendx	⊢ ( ·𝑠 ‘ ndx ) ≠ ( Base ‘ ndx )
5	1 2 3 4	resseqnbas	⊢ ( 𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘ 𝐻 ) )