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Mirrors > Home > MPE Home > Th. List > unirnbl | Structured version Visualization version 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 22212 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | |
2 | frn 6053 | . . . 4 ⊢ ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 → ran (ball‘𝐷) ⊆ 𝒫 𝑋) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ⊆ 𝒫 𝑋) |
4 | sspwuni 4611 | . . 3 ⊢ (ran (ball‘𝐷) ⊆ 𝒫 𝑋 ↔ ∪ ran (ball‘𝐷) ⊆ 𝑋) | |
5 | 3, 4 | sylib 208 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) ⊆ 𝑋) |
6 | 1rp 11836 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
7 | blcntr 22218 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐷)1)) | |
8 | 6, 7 | mp3an3 1413 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (𝑥(ball‘𝐷)1)) |
9 | rpxr 11840 | . . . . . . 7 ⊢ (1 ∈ ℝ+ → 1 ∈ ℝ*) | |
10 | 6, 9 | ax-mp 5 | . . . . . 6 ⊢ 1 ∈ ℝ* |
11 | blelrn 22222 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ*) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) | |
12 | 10, 11 | mp3an3 1413 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) |
13 | elunii 4441 | . . . . 5 ⊢ ((𝑥 ∈ (𝑥(ball‘𝐷)1) ∧ (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) → 𝑥 ∈ ∪ ran (ball‘𝐷)) | |
14 | 8, 12, 13 | syl2anc 693 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ ran (ball‘𝐷)) |
15 | 14 | ex 450 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝑋 → 𝑥 ∈ ∪ ran (ball‘𝐷))) |
16 | 15 | ssrdv 3609 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ⊆ ∪ ran (ball‘𝐷)) |
17 | 5, 16 | eqssd 3620 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 × cxp 5112 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1c1 9937 ℝ*cxr 10073 ℝ+crp 11832 ∞Metcxmt 19731 ballcbl 19733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-rp 11833 df-psmet 19738 df-xmet 19739 df-bl 19741 |
This theorem is referenced by: blbas 22235 mopntopon 22244 elmopn 22247 imasf1oxms 22294 metss 22313 |
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