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Theorem blrnps 24332
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 24330 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
2 ffn 6725 . 2 ((ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋 β†’ (ballβ€˜π·) Fn (𝑋 Γ— ℝ*))
3 ovelrn 7601 . 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 18 1 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐴 ∈ ran (ballβ€˜π·) ↔ βˆƒπ‘₯ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ* 𝐴 = (π‘₯(ballβ€˜π·)π‘Ÿ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3066  π’« cpw 4604   Γ— cxp 5678  ran crn 5681   Fn wfn 6546  βŸΆwf 6547  β€˜cfv 6551  (class class class)co 7424  β„*cxr 11283  PsMetcpsmet 21268  ballcbl 21271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-map 8851  df-xr 11288  df-psmet 21276  df-bl 21279
This theorem is referenced by:  blssps  24348
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