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Theorem unirnblps 22164
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 22151 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
2 frn 6020 . . . 4 ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
31, 2syl 17 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
4 sspwuni 4584 . . 3 (ran (ball‘𝐷) ⊆ 𝒫 𝑋 ran (ball‘𝐷) ⊆ 𝑋)
53, 4sylib 208 . 2 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) ⊆ 𝑋)
6 1rp 11796 . . . . . 6 1 ∈ ℝ+
7 blcntrps 22157 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋 ∧ 1 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐷)1))
86, 7mp3an3 1410 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → 𝑥 ∈ (𝑥(ball‘𝐷)1))
9 rpxr 11800 . . . . . . 7 (1 ∈ ℝ+ → 1 ∈ ℝ*)
106, 9ax-mp 5 . . . . . 6 1 ∈ ℝ*
11 blelrnps 22161 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋 ∧ 1 ∈ ℝ*) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷))
1210, 11mp3an3 1410 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷))
13 elunii 4414 . . . . 5 ((𝑥 ∈ (𝑥(ball‘𝐷)1) ∧ (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) → 𝑥 ran (ball‘𝐷))
148, 12, 13syl2anc 692 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → 𝑥 ran (ball‘𝐷))
1514ex 450 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋𝑥 ran (ball‘𝐷)))
1615ssrdv 3594 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ran (ball‘𝐷))
175, 16eqssd 3605 1 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3560  𝒫 cpw 4136   cuni 4409   × cxp 5082  ran crn 5085  wf 5853  cfv 5857  (class class class)co 6615  1c1 9897  *cxr 10033  +crp 11792  PsMetcpsmet 19670  ballcbl 19673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-rp 11793  df-psmet 19678  df-bl 19681
This theorem is referenced by:  psmetutop  22312
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