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Mirrors > Home > ILE Home > Th. List > unirnbl | GIF version |
Description: The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
unirnbl | ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blf 13846 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | |
2 | 1 | frnd 5375 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ⊆ 𝒫 𝑋) |
3 | sspwuni 3971 | . . 3 ⊢ (ran (ball‘𝐷) ⊆ 𝒫 𝑋 ↔ ∪ ran (ball‘𝐷) ⊆ 𝑋) | |
4 | 2, 3 | sylib 122 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) ⊆ 𝑋) |
5 | 1rp 9656 | . . . 4 ⊢ 1 ∈ ℝ+ | |
6 | blcntr 13852 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐷)1)) | |
7 | 5, 6 | mp3an3 1326 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (𝑥(ball‘𝐷)1)) |
8 | rpxr 9660 | . . . . 5 ⊢ (1 ∈ ℝ+ → 1 ∈ ℝ*) | |
9 | 5, 8 | ax-mp 5 | . . . 4 ⊢ 1 ∈ ℝ* |
10 | blelrn 13856 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ*) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) | |
11 | 9, 10 | mp3an3 1326 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) |
12 | elunii 3814 | . . 3 ⊢ ((𝑥 ∈ (𝑥(ball‘𝐷)1) ∧ (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) → 𝑥 ∈ ∪ ran (ball‘𝐷)) | |
13 | 7, 11, 12 | syl2anc 411 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ ran (ball‘𝐷)) |
14 | 4, 13 | eqelssd 3174 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ⊆ wss 3129 𝒫 cpw 3575 ∪ cuni 3809 × cxp 4624 ran crn 4627 ‘cfv 5216 (class class class)co 5874 1c1 7811 ℝ*cxr 7990 ℝ+crp 9652 ∞Metcxmet 13376 ballcbl 13378 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 ax-0lt1 7916 ax-rnegex 7919 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-map 6649 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-rp 9653 df-psmet 13383 df-xmet 13384 df-bl 13386 |
This theorem is referenced by: blbas 13869 mopntopon 13879 elmopn 13882 metss 13930 xmettx 13946 |
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