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Theorem blbas 24440
Description: The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)
Assertion
Ref Expression
blbas (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)

Proof of Theorem blbas
Dummy variables 𝑥 𝑟 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blin2 24439 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ (𝑥𝑦))
2 simpll 767 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝐷 ∈ (∞Met‘𝑋))
3 elinel1 4201 . . . . . . . . . 10 (𝑧 ∈ (𝑥𝑦) → 𝑧𝑥)
4 elunii 4912 . . . . . . . . . 10 ((𝑧𝑥𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ran (ball‘𝐷))
53, 4sylan 580 . . . . . . . . 9 ((𝑧 ∈ (𝑥𝑦) ∧ 𝑥 ∈ ran (ball‘𝐷)) → 𝑧 ran (ball‘𝐷))
65ad2ant2lr 748 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧 ran (ball‘𝐷))
7 unirnbl 24430 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) = 𝑋)
87ad2antrr 726 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ran (ball‘𝐷) = 𝑋)
96, 8eleqtrd 2843 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → 𝑧𝑋)
10 blssex 24437 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧𝑋) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)) ↔ ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ (𝑥𝑦)))
112, 9, 10syl2anc 584 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)) ↔ ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ (𝑥𝑦)))
121, 11mpbird 257 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) ∧ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷))) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)))
1312ex 412 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ (𝑥𝑦)) → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷)) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦))))
1413ralrimdva 3154 . . 3 (𝐷 ∈ (∞Met‘𝑋) → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑦 ∈ ran (ball‘𝐷)) → ∀𝑧 ∈ (𝑥𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦))))
1514ralrimivv 3200 . 2 (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)))
16 fvex 6919 . . . 4 (ball‘𝐷) ∈ V
1716rnex 7932 . . 3 ran (ball‘𝐷) ∈ V
18 isbasis2g 22955 . . 3 (ran (ball‘𝐷) ∈ V → (ran (ball‘𝐷) ∈ TopBases ↔ ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦))))
1917, 18ax-mp 5 . 2 (ran (ball‘𝐷) ∈ TopBases ↔ ∀𝑥 ∈ ran (ball‘𝐷)∀𝑦 ∈ ran (ball‘𝐷)∀𝑧 ∈ (𝑥𝑦)∃𝑏 ∈ ran (ball‘𝐷)(𝑧𝑏𝑏 ⊆ (𝑥𝑦)))
2015, 19sylibr 234 1 (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951   cuni 4907  ran crn 5686  cfv 6561  (class class class)co 7431  +crp 13034  ∞Metcxmet 21349  ballcbl 21351  TopBasesctb 22952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-psmet 21356  df-xmet 21357  df-bl 21359  df-bases 22953
This theorem is referenced by:  mopntopon  24449  elmopn  24452  imasf1oxms  24502  blssopn  24508  metss  24521
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