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| Mirrors > Home > ILE Home > Th. List > blex | GIF version | ||
| Description: A ball is a set. Also see blfn 14568 in case you just know 𝐷 is a set, not 𝐷 ∈ (∞Met‘𝑋). (Contributed by Jim Kingdon, 4-May-2023.) |
| Ref | Expression |
|---|---|
| blex | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | blfval 15113 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})) | |
| 2 | xmetrel 15070 | . . . 4 ⊢ Rel ∞Met | |
| 3 | relelfvdm 5671 | . . . 4 ⊢ ((Rel ∞Met ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝑋 ∈ dom ∞Met) | |
| 4 | 2, 3 | mpan 424 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) |
| 5 | xrex 10091 | . . 3 ⊢ ℝ* ∈ V | |
| 6 | mpoexga 6377 | . . 3 ⊢ ((𝑋 ∈ dom ∞Met ∧ ℝ* ∈ V) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V) | |
| 7 | 4, 5, 6 | sylancl 413 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V) |
| 8 | 1, 7 | eqeltrd 2308 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) ∈ V) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∈ wcel 2202 {crab 2514 Vcvv 2802 class class class wbr 4088 dom cdm 4725 Rel wrel 4730 ‘cfv 5326 (class class class)co 6018 ∈ cmpo 6020 ℝ*cxr 8213 < clt 8214 ∞Metcxmet 14553 ballcbl 14555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 |
| This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-map 6819 df-pnf 8216 df-mnf 8217 df-xr 8218 df-psmet 14560 df-xmet 14561 df-bl 14563 |
| This theorem is referenced by: blbas 15160 metrest 15233 xmettxlem 15236 xmettx 15237 tgioo 15281 |
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