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Theorem ballss3 42643
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 483 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3 simpr 485 . . . . . . 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 23540 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
84, 5, 6, 7syl3anc 1370 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
98adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
103, 9mpbid 231 . . . . . 6 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))
1110simpld 495 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝑋)
1210simprd 496 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
13 ballss3.a . . . . 5 ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)
142, 11, 12, 13syl3anc 1370 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝐴)
1514ex 413 . . 3 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥𝐴))
161, 15ralrimi 3141 . 2 (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥𝐴)
17 dfss3 3909 . 2 ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥𝐴)
1816, 17sylibr 233 1 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wnf 1786  wcel 2106  wral 3064  wss 3887   class class class wbr 5074  cfv 6433  (class class class)co 7275  *cxr 11008   < clt 11009  PsMetcpsmet 20581  ballcbl 20584
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-xr 11013  df-psmet 20589  df-bl 20592
This theorem is referenced by:  ioorrnopnlem  43845
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