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Theorem unirnblps 14325
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.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
unirnblps (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ βˆͺ ran (ballβ€˜π·) = 𝑋)

Proof of Theorem unirnblps
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 blfps 14312 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
21frnd 5390 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ran (ballβ€˜π·) βŠ† 𝒫 𝑋)
3 sspwuni 3986 . . 3 (ran (ballβ€˜π·) βŠ† 𝒫 𝑋 ↔ βˆͺ ran (ballβ€˜π·) βŠ† 𝑋)
42, 3sylib 122 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ βˆͺ ran (ballβ€˜π·) βŠ† 𝑋)
5 1rp 9676 . . . 4 1 ∈ ℝ+
6 blcntrps 14318 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋 ∧ 1 ∈ ℝ+) β†’ π‘₯ ∈ (π‘₯(ballβ€˜π·)1))
75, 6mp3an3 1337 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ (π‘₯(ballβ€˜π·)1))
8 rpxr 9680 . . . . 5 (1 ∈ ℝ+ β†’ 1 ∈ ℝ*)
95, 8ax-mp 5 . . . 4 1 ∈ ℝ*
10 blelrnps 14322 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋 ∧ 1 ∈ ℝ*) β†’ (π‘₯(ballβ€˜π·)1) ∈ ran (ballβ€˜π·))
119, 10mp3an3 1337 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯(ballβ€˜π·)1) ∈ ran (ballβ€˜π·))
12 elunii 3829 . . 3 ((π‘₯ ∈ (π‘₯(ballβ€˜π·)1) ∧ (π‘₯(ballβ€˜π·)1) ∈ ran (ballβ€˜π·)) β†’ π‘₯ ∈ βˆͺ ran (ballβ€˜π·))
137, 11, 12syl2anc 411 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ βˆͺ ran (ballβ€˜π·))
144, 13eqelssd 3189 1 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ βˆͺ ran (ballβ€˜π·) = 𝑋)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160   βŠ† wss 3144  π’« cpw 3590  βˆͺ cuni 3824   Γ— cxp 4639  ran crn 4642  β€˜cfv 5231  (class class class)co 5891  1c1 7831  β„*cxr 8010  β„+crp 9672  PsMetcpsmet 13815  ballcbl 13818
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927  ax-0lt1 7936  ax-rnegex 7939
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-map 6668  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-rp 9673  df-psmet 13823  df-bl 13826
This theorem is referenced by: (None)
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