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Theorem metdsval 22697
 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))
Assertion
Ref Expression
metdsval (𝐴𝑋 → (𝐹𝐴) = inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
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
StepHypRef Expression
1 oveq1 6697 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
21mpteq2dv 4778 . . . 4 (𝑥 = 𝐴 → (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
32rneqd 5385 . . 3 (𝑥 = 𝐴 → ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
43infeq1d 8424 . 2 (𝑥 = 𝐴 → inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) = inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
5 metdscn.f . 2 𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))
6 xrltso 12012 . . 3 < Or ℝ*
76infex 8440 . 2 inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ∈ V
84, 5, 7fvmpt 6321 1 (𝐴𝑋 → (𝐹𝐴) = inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030   ↦ cmpt 4762  ran crn 5144  ‘cfv 5926  (class class class)co 6690  infcinf 8388  ℝ*cxr 10111   < clt 10112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117 This theorem is referenced by:  metdsge  22699  lebnumlem1  22807  lebnumlem3  22809
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