Theorem metdsval 22650

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.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x, y, A   x, D, y   x, S, y   x, X, y

Allowed substitution hints:   F( x, y)

Proof of Theorem metdsval

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 |- ( x = A -> ( x D y ) = ( A D y ) )
2	1	mpteq2dv 4745	. . . 4 |- ( x = A -> ( y e. S |-> ( x D y ) ) = ( y e. S |-> ( A D y ) ) )
3	2	rneqd 5353	. . 3 |- ( x = A -> ran ( y e. S |-> ( x D y ) ) = ran ( y e. S |-> ( A D y ) ) )
4	3	infeq1d 8383	. 2 |- ( x = A -> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) = inf ( ran ( y e. S |-> ( A D y ) ) , RR* , < ) )
5		metdscn.f	. 2 |- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) )
6		xrltso 11974	. . 3 |- < Or RR*
7	6	infex 8399	. 2 |- inf ( ran ( y e. S |-> ( A D y ) ) , RR* , < ) e. _V
8	4, 5, 7	fvmpt 6282	1 |- ( A e. X -> ( F ` A ) = inf ( ran ( y e. S |-> ( A D y ) ) , RR* , < ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650  infcinf 8347   RR*cxr 10073   < clt 10074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079

This theorem is referenced by:  metdsge  22652  lebnumlem1  22760  lebnumlem3  22762