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Mirrors > Home > ILE Home > Th. List > blelrn | GIF version |
Description: A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
blelrn | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blf 12618 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | |
2 | 1 | ffnd 5281 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) Fn (𝑋 × ℝ*)) |
3 | fnovrn 5926 | . 2 ⊢ (((ball‘𝐷) Fn (𝑋 × ℝ*) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷)) | |
4 | 2, 3 | syl3an1 1250 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 963 ∈ wcel 1481 𝒫 cpw 3515 × cxp 4545 ran crn 4548 Fn wfn 5126 ‘cfv 5131 (class class class)co 5782 ℝ*cxr 7823 ∞Metcxmet 12188 ballcbl 12190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-map 6552 df-pnf 7826 df-mnf 7827 df-xr 7828 df-psmet 12195 df-xmet 12196 df-bl 12198 |
This theorem is referenced by: unirnbl 12631 blssex 12638 blopn 12698 metss 12702 metcnp3 12719 ioo2blex 12752 |
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