Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballss3 Structured version   Visualization version   GIF version

Theorem ballss3 41651
Description: A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
ballss3.y 𝑥𝜑
ballss3.d (𝜑𝐷 ∈ (PsMet‘𝑋))
ballss3.p (𝜑𝑃𝑋)
ballss3.r (𝜑𝑅 ∈ ℝ*)
ballss3.a ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)
Assertion
Ref Expression
ballss3 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐷   𝑥,𝑃   𝑥,𝑅
Allowed substitution hints:   𝜑(𝑥)   𝑋(𝑥)

Proof of Theorem ballss3
StepHypRef Expression
1 ballss3.y . . 3 𝑥𝜑
2 simpl 486 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3 simpr 488 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
4 ballss3.d . . . . . . . . 9 (𝜑𝐷 ∈ (PsMet‘𝑋))
5 ballss3.p . . . . . . . . 9 (𝜑𝑃𝑋)
6 ballss3.r . . . . . . . . 9 (𝜑𝑅 ∈ ℝ*)
7 elblps 22997 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
84, 5, 6, 7syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
98adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
103, 9mpbid 235 . . . . . 6 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))
1110simpld 498 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝑋)
1210simprd 499 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
13 ballss3.a . . . . 5 ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)
142, 11, 12, 13syl3anc 1368 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝐴)
1514ex 416 . . 3 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥𝐴))
161, 15ralrimi 3210 . 2 (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥𝐴)
17 dfss3 3941 . 2 ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥𝐴)
1816, 17sylibr 237 1 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wnf 1785  wcel 2115  wral 3133  wss 3919   class class class wbr 5052  cfv 6343  (class class class)co 7149  *cxr 10672   < clt 10673  PsMetcpsmet 20529  ballcbl 20532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-xr 10677  df-psmet 20537  df-bl 20540
This theorem is referenced by:  ioorrnopnlem  42872
  Copyright terms: Public domain W3C validator