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		df-ds	⊢ dist = Slot ; 1 2
4		1nn0	⊢ 1 ∈ ℕ0
5		2nn	⊢ 2 ∈ ℕ
6	4 5	decnncl	⊢ ; 1 2 ∈ ℕ
7		1nn	⊢ 1 ∈ ℕ
8		2nn0	⊢ 2 ∈ ℕ0
9		1lt10	⊢ 1 < ; 1 0
10	7 8 4 9	declti	⊢ 1 < ; 1 2
11	1 2 3 6 10	resslem	⊢ ( 𝐴 ∈ 𝑉 → 𝐷 = ( dist ‘ 𝐻 ) )