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Theorem blssex 24326
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 24324 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ π‘₯) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯)
2 sstr 3986 . . . . . . . . 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 3165 . . . . . . 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 1116 . . . . 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 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 3150 . 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 13009 . . . . . 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 24316 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·))
1510, 11, 13, 14syl3anc 1369 . . . 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 24312 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ))
1810, 11, 16, 17syl3anc 1369 . . . 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 2817 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ)))
21 sseq1 4003 . . . . . 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 3607 . . . 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 3151 . 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 1085   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3065   βŠ† wss 3944  ran crn 5673  β€˜cfv 6542  (class class class)co 7414  β„*cxr 11271  β„+crp 13000  βˆžMetcxmet 21257  ballcbl 21259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9459  df-inf 9460  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-n0 12497  df-z 12583  df-uz 12847  df-q 12957  df-rp 13001  df-xneg 13118  df-xadd 13119  df-xmul 13120  df-psmet 21264  df-xmet 21265  df-bl 21267
This theorem is referenced by:  blbas  24329  elmopn2  24344  mopni2  24395  metss  24410  tgioo  24705
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