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Theorem metdsval 24226
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 7365 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
21mpteq2dv 5208 . . . 4 (𝑥 = 𝐴 → (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
32rneqd 5894 . . 3 (𝑥 = 𝐴 → ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)) = ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)))
43infeq1d 9418 . 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 13066 . . 3 < Or ℝ*
76infex 9434 . 2 inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ∈ V
84, 5, 7fvmpt 6949 1 (𝐴𝑋 → (𝐹𝐴) = inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cmpt 5189  ran crn 5635  cfv 6497  (class class class)co 7358  infcinf 9382  *cxr 11193   < clt 11194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-pre-lttri 11130  ax-pre-lttrn 11131
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-inf 9384  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199
This theorem is referenced by:  metdsge  24228  lebnumlem1  24340  lebnumlem3  24342
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