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Theorem blssex 24255
Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
blssex ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
Distinct variable groups:   π‘₯,π‘Ÿ,𝐴   𝐷,π‘Ÿ,π‘₯   𝑃,π‘Ÿ,π‘₯   𝑋,π‘Ÿ,π‘₯
Proof of Theorem blssex
StepHypRef Expression
1 blss 24253 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ π‘₯) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯)
2 sstr 3982 . . . . . . . . 9 (((𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)
32expcom 413 . . . . . . . 8 (π‘₯ βŠ† 𝐴 β†’ ((𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
43reximdv 3162 . . . . . . 7 (π‘₯ βŠ† 𝐴 β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
51, 4syl5com 31 . . . . . 6 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ π‘₯) β†’ (π‘₯ βŠ† 𝐴 β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
653expa 1115 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·)) ∧ 𝑃 ∈ π‘₯) β†’ (π‘₯ βŠ† 𝐴 β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
76expimpd 453 . . . 4 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·)) β†’ ((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
87adantlr 712 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ π‘₯ ∈ ran (ballβ€˜π·)) β†’ ((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
98rexlimdva 3147 . 2 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
10 simpll 764 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
11 simplr 766 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝑃 ∈ 𝑋)
12 rpxr 12980 . . . . . 6 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
1312ad2antrl 725 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ π‘Ÿ ∈ ℝ*)
14 blelrn 24245 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·))
1510, 11, 13, 14syl3anc 1368 . . . 4 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·))
16 simprl 768 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ π‘Ÿ ∈ ℝ+)
17 blcntr 24241 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ))
1810, 11, 16, 17syl3anc 1368 . . . 4 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ))
19 simprr 770 . . . 4 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)
20 eleq2 2814 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ)))
21 sseq1 3999 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (π‘₯ βŠ† 𝐴 ↔ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
2220, 21anbi12d 630 . . . . 5 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ ((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ (𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ) ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)))
2322rspcev 3604 . . . 4 (((𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·) ∧ (𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ) ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴))
2415, 18, 19, 23syl12anc 834 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴))
2524rexlimdvaa 3148 . 2 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴 β†’ βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
269, 25impbid 211 1 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062   βŠ† wss 3940  ran crn 5667  β€˜cfv 6533  (class class class)co 7401  β„*cxr 11244  β„+crp 12971  βˆžMetcxmet 21213  ballcbl 21215
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 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-psmet 21220  df-xmet 21221  df-bl 21223
This theorem is referenced by:  blbas  24258  elmopn2  24273  mopni2  24324  metss  24339  tgioo  24634
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