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 e. 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 e. NN0
5		2nn	|- 2 e. NN
6	4 5	decnncl	|- ; 1 2 e. NN
7		1nn	|- 1 e. NN
8		2nn0	|- 2 e. NN0
9		1lt10	|- 1 < ; 1 0
10	7 8 4 9	declti	|- 1 < ; 1 2
11	1 2 3 6 10	resslem	|- ( A e. V -> D = ( dist ` H ) )