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* , < ) )