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Theorem ballss3 38788
 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 473 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3 simpr 477 . . . . . . 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 22115 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
84, 5, 6, 7syl3anc 1323 . . . . . . . 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 222 . . . . . 6 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))
1110simpld 475 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝑋)
1210simprd 479 . . . . 5 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
13 ballss3.a . . . . 5 ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)
142, 11, 12, 13syl3anc 1323 . . . 4 ((𝜑𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥𝐴)
1514ex 450 . . 3 (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥𝐴))
161, 15ralrimi 2952 . 2 (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥𝐴)
17 dfss3 3577 . 2 ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥𝐴)
1816, 17sylibr 224 1 (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036  Ⅎwnf 1705   ∈ wcel 1987  ∀wral 2907   ⊆ wss 3559   class class class wbr 4618  ‘cfv 5852  (class class class)co 6610  ℝ*cxr 10025   < clt 10026  PsMetcpsmet 19662  ballcbl 19665 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-map 7811  df-xr 10030  df-psmet 19670  df-bl 19673 This theorem is referenced by:  ioorrnopnlem  39857
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