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Metamath Proof Explorer

Theorem metdsval

Description: Value of the "distance to a set" function. (Contributed by Mario Carneiro, 14-Feb-2015) (Revised by Mario Carneiro, 4-Sep-2015) (Revised by AV, 30-Sep-2020)

		Ref	Expression
	Hypothesis	metdscn.f	\|- F = ( x e. X \|-> inf ( ran ( y e. S \|-> ( x D y ) ) , RR* , < ) )
	Assertion	metdsval	\|- ( A e. X -> ( F ` A ) = inf ( ran ( y e. S \|-> ( A D y ) ) , RR* , < ) )

Proof

Step	Hyp	Ref	Expression
1		metdscn.f	\|- F = ( x e. X \|-> inf ( ran ( y e. S \|-> ( x D y ) ) , RR* , < ) )
2		oveq1	\|- ( x = A -> ( x D y ) = ( A D y ) )
3	2	mpteq2dv	\|- ( x = A -> ( y e. S \|-> ( x D y ) ) = ( y e. S \|-> ( A D y ) ) )
4	3	rneqd	\|- ( x = A -> ran ( y e. S \|-> ( x D y ) ) = ran ( y e. S \|-> ( A D y ) ) )
5	4	infeq1d	\|- ( x = A -> inf ( ran ( y e. S \|-> ( x D y ) ) , RR* , < ) = inf ( ran ( y e. S \|-> ( A D y ) ) , RR* , < ) )
6		xrltso	\|- < Or RR*
7	6	infex	\|- inf ( ran ( y e. S \|-> ( A D y ) ) , RR* , < ) e. _V
8	5 1 7	fvmpt	\|- ( A e. X -> ( F ` A ) = inf ( ran ( y e. S \|-> ( A D y ) ) , RR* , < ) )