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| Mirrors > Home > MPE Home > Th. List > metdsval | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| metdscn.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) |
| Ref | Expression |
|---|---|
| metdsval | ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦)) | |
| 2 | 1 | mpteq2dv 5244 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦))) |
| 3 | 2 | rneqd 5949 | . . 3 ⊢ (𝑥 = 𝐴 → ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) = ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦))) |
| 4 | 3 | infeq1d 9517 | . 2 ⊢ (𝑥 = 𝐴 → inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < )) |
| 5 | metdscn.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
| 6 | xrltso 13183 | . . 3 ⊢ < Or ℝ* | |
| 7 | 6 | infex 9533 | . 2 ⊢ inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ∈ V |
| 8 | 4, 5, 7 | fvmpt 7016 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 infcinf 9481 ℝ*cxr 11294 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 |
| This theorem is referenced by: metdsge 24871 lebnumlem1 24993 lebnumlem3 24995 |
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