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Theorem metdsval 24783
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 7433 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
21mpteq2dv 5254 . . . 4 (𝑥 = 𝐴 → (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
32rneqd 5944 . . 3 (𝑥 = 𝐴 → ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
43infeq1d 9508 . 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 13160 . . 3 < Or ℝ*
76infex 9524 . 2 inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ∈ V
84, 5, 7fvmpt 7010 1 (𝐴𝑋 → (𝐹𝐴) = inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cmpt 5235  ran crn 5683  cfv 6553  (class class class)co 7426  infcinf 9472  *cxr 11285   < clt 11286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-pre-lttri 11220  ax-pre-lttrn 11221
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-sup 9473  df-inf 9474  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291
This theorem is referenced by:  metdsge  24785  lebnumlem1  24907  lebnumlem3  24909
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