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Mirrors > Home > MPE Home > Th. List > blf | Structured version Visualization version GIF version |
Description: Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
blf | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3914 | . . . . . 6 ⊢ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋 | |
2 | elfvdm 6469 | . . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
3 | elpw2g 5051 | . . . . . . 7 ⊢ (𝑋 ∈ dom ∞Met → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋)) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋)) |
5 | 1, 4 | mpbiri 250 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋) |
6 | 5 | a1d 25 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)) |
7 | 6 | ralrimivv 3179 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋) |
8 | eqid 2825 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) | |
9 | 8 | fmpt2 7505 | . . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋) |
10 | 7, 9 | sylib 210 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋) |
11 | blfval 22566 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})) | |
12 | 11 | feq1d 6267 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)) |
13 | 10, 12 | mpbird 249 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 ∀wral 3117 {crab 3121 ⊆ wss 3798 𝒫 cpw 4380 class class class wbr 4875 × cxp 5344 dom cdm 5346 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 ↦ cmpt2 6912 ℝ*cxr 10397 < clt 10398 ∞Metcxmet 20098 ballcbl 20100 |
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 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 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-ral 3122 df-rex 3123 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-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 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-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-map 8129 df-xr 10402 df-psmet 20105 df-xmet 20106 df-bl 20108 |
This theorem is referenced by: blrn 22591 blelrn 22599 blssm 22600 unirnbl 22602 blin2 22611 imasf1oxms 22671 iscau2 23452 ismtyhmeolem 34140 |
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