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Theorem metdsval 23027
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 6917 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
21mpteq2dv 4970 . . . 4 (𝑥 = 𝐴 → (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
32rneqd 5589 . . 3 (𝑥 = 𝐴 → ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
43infeq1d 8658 . 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 12267 . . 3 < Or ℝ*
76infex 8674 . 2 inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ∈ V
84, 5, 7fvmpt 6533 1 (𝐴𝑋 → (𝐹𝐴) = inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  cmpt 4954  ran crn 5347  cfv 6127  (class class class)co 6910  infcinf 8622  *cxr 10397   < clt 10398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-pre-lttri 10333  ax-pre-lttrn 10334
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-po 5265  df-so 5266  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-sup 8623  df-inf 8624  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403
This theorem is referenced by:  metdsge  23029  lebnumlem1  23137  lebnumlem3  23139
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