Theorem ressds

Description: dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015)

	Ref	Expression
Hypotheses	ressds.1	⊢ 𝐻 = ( 𝐺 ↾s 𝐴 )
	ressds.2	⊢ 𝐷 = ( dist ‘ 𝐺 )
Assertion	ressds	⊢ ( 𝐴 ∈ 𝑉 → 𝐷 = ( dist ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		ressds.1	⊢ 𝐻 = ( 𝐺 ↾s 𝐴 )
2		ressds.2	⊢ 𝐷 = ( dist ‘ 𝐺 )
3		dsid	⊢ dist = Slot ( dist ‘ ndx )
4		dsndxnbasendx	⊢ ( dist ‘ ndx ) ≠ ( Base ‘ ndx )
5	1 2 3 4	resseqnbas	⊢ ( 𝐴 ∈ 𝑉 → 𝐷 = ( dist ‘ 𝐻 ) )