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Theorem unirnblps 13582
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 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) = 𝑋)

Proof of Theorem unirnblps
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 blfps 13569 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
21frnd 5371 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
3 sspwuni 3968 . . 3 (ran (ball‘𝐷) ⊆ 𝒫 𝑋 ran (ball‘𝐷) ⊆ 𝑋)
42, 3sylib 122 . 2 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) ⊆ 𝑋)
5 1rp 9641 . . . 4 1 ∈ ℝ+
6 blcntrps 13575 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋 ∧ 1 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐷)1))
75, 6mp3an3 1326 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → 𝑥 ∈ (𝑥(ball‘𝐷)1))
8 rpxr 9645 . . . . 5 (1 ∈ ℝ+ → 1 ∈ ℝ*)
95, 8ax-mp 5 . . . 4 1 ∈ ℝ*
10 blelrnps 13579 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋 ∧ 1 ∈ ℝ*) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷))
119, 10mp3an3 1326 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷))
12 elunii 3812 . . 3 ((𝑥 ∈ (𝑥(ball‘𝐷)1) ∧ (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) → 𝑥 ran (ball‘𝐷))
137, 11, 12syl2anc 411 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → 𝑥 ran (ball‘𝐷))
144, 13eqelssd 3174 1 (𝐷 ∈ (PsMet‘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 3574   cuni 3807   × cxp 4621  ran crn 4624  cfv 5212  (class class class)co 5869  1c1 7800  *cxr 7978  +crp 9637  PsMetcpsmet 13139  ballcbl 13142
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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1re 7893  ax-addrcl 7896  ax-0lt1 7905  ax-rnegex 7908
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-pnf 7981  df-mnf 7982  df-xr 7983  df-ltxr 7984  df-rp 9638  df-psmet 13147  df-bl 13150
This theorem is referenced by: (None)
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