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	$$⊢F=\left(x\in X⟼sup\left(\mathrm{ran}\left(y\in S⟼xDy\right),{ℝ}^{*},<\right)\right)$$
Assertion	metdsval	$$⊢A\in X\to F\left(A\right)=sup\left(\mathrm{ran}\left(y\in S⟼ADy\right),{ℝ}^{*},<\right)$$

Proof

Step	Hyp	Ref	Expression
1		metdscn.f	$$⊢F=\left(x\in X⟼sup\left(\mathrm{ran}\left(y\in S⟼xDy\right),{ℝ}^{*},<\right)\right)$$
2		oveq1	$$⊢x=A\to xDy=ADy$$
3	2	mpteq2dv	$$⊢x=A\to \left(y\in S⟼xDy\right)=\left(y\in S⟼ADy\right)$$
4	3	rneqd	$$⊢x=A\to \mathrm{ran}\left(y\in S⟼xDy\right)=\mathrm{ran}\left(y\in S⟼ADy\right)$$
5	4	infeq1d	$$⊢x=A\to sup\left(\mathrm{ran}\left(y\in S⟼xDy\right),{ℝ}^{*},<\right)=sup\left(\mathrm{ran}\left(y\in S⟼ADy\right),{ℝ}^{*},<\right)$$
6		xrltso	$$⊢<\mathrm{Or}{ℝ}^{*}$$
7	6	infex	$$⊢sup\left(\mathrm{ran}\left(y\in S⟼ADy\right),{ℝ}^{*},<\right)\in \mathrm{V}$$
8	5 1 7	fvmpt	$$⊢A\in X\to F\left(A\right)=sup\left(\mathrm{ran}\left(y\in S⟼ADy\right),{ℝ}^{*},<\right)$$