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Theorem blssex 23933
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 23931 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ π‘₯) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯)
2 sstr 3991 . . . . . . . . 9 (((𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)
32expcom 415 . . . . . . . 8 (π‘₯ βŠ† 𝐴 β†’ ((𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
43reximdv 3171 . . . . . . 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 1119 . . . . 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 455 . . . 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 714 . . 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 3156 . 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 766 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
11 simplr 768 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝑃 ∈ 𝑋)
12 rpxr 12983 . . . . . 6 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
1312ad2antrl 727 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ π‘Ÿ ∈ ℝ*)
14 blelrn 23923 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·))
1510, 11, 13, 14syl3anc 1372 . . . 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 770 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ π‘Ÿ ∈ ℝ+)
17 blcntr 23919 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ))
1810, 11, 16, 17syl3anc 1372 . . . 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 772 . . . 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 2823 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ)))
21 sseq1 4008 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (π‘₯ βŠ† 𝐴 ↔ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
2220, 21anbi12d 632 . . . . 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 3613 . . . 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 836 . . 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 3157 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   βŠ† wss 3949  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  β„*cxr 11247  β„+crp 12974  βˆžMetcxmet 20929  ballcbl 20931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-psmet 20936  df-xmet 20937  df-bl 20939
This theorem is referenced by:  blbas  23936  elmopn2  23951  mopni2  24002  metss  24017  tgioo  24312
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