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Theorem blrnps 24265
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 24263 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
2 ffn 6710 . 2 ((ballβ€˜π·):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋 β†’ (ballβ€˜π·) Fn (𝑋 Γ— ℝ*))
3 ovelrn 7579 . 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 3064  π’« cpw 4597   Γ— cxp 5667  ran crn 5670   Fn wfn 6531  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  β„*cxr 11248  PsMetcpsmet 21220  ballcbl 21223
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-xr 11253  df-psmet 21228  df-bl 21231
This theorem is referenced by:  blssps  24281
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