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Theorem metdsval 24883
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 7438 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
21mpteq2dv 5250 . . . 4 (𝑥 = 𝐴 → (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
32rneqd 5952 . . 3 (𝑥 = 𝐴 → ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
43infeq1d 9515 . 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 13180 . . 3 < Or ℝ*
76infex 9531 . 2 inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ∈ V
84, 5, 7fvmpt 7016 1 (𝐴𝑋 → (𝐹𝐴) = inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  infcinf 9479  *cxr 11292   < clt 11293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298
This theorem is referenced by:  metdsge  24885  lebnumlem1  25007  lebnumlem3  25009
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