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Theorem blrnps 14295
Description: Membership in the range of the ball function. Note that ran (ballβ€˜π·) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blrnps (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐴 ∈ ran (ballβ€˜π·) ↔ βˆƒπ‘₯ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ* 𝐴 = (π‘₯(ballβ€˜π·)π‘Ÿ)))
Distinct variable groups:   π‘₯,π‘Ÿ,𝐴   𝐷,π‘Ÿ,π‘₯   𝑋,π‘Ÿ,π‘₯

Proof of Theorem blrnps
StepHypRef Expression
1 blfps 14293 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
2 ffn 5380 . 2 ((ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋 β†’ (ballβ€˜π·) Fn (𝑋 Γ— ℝ*))
3 ovelrn 6040 . 2 ((ballβ€˜π·) Fn (𝑋 Γ— ℝ*) β†’ (𝐴 ∈ ran (ballβ€˜π·) ↔ βˆƒπ‘₯ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ* 𝐴 = (π‘₯(ballβ€˜π·)π‘Ÿ)))
41, 2, 33syl 17 1 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐴 ∈ ran (ballβ€˜π·) ↔ βˆƒπ‘₯ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ* 𝐴 = (π‘₯(ballβ€˜π·)π‘Ÿ)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2160  βˆƒwrex 2469  π’« cpw 3590   Γ— cxp 4639  ran crn 4642   Fn wfn 5226  βŸΆwf 5227  β€˜cfv 5231  (class class class)co 5891  β„*cxr 8009  PsMetcpsmet 13809  ballcbl 13812
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 7920  ax-resscn 7921
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-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 8012  df-mnf 8013  df-xr 8014  df-psmet 13817  df-bl 13820
This theorem is referenced by:  blssps  14311
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