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Theorem isbnd 34939
Description: The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
isbnd (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
Distinct variable groups:   𝑥,𝑟,𝑀   𝑋,𝑟,𝑥

Proof of Theorem isbnd
Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6696 . 2 (𝑀 ∈ (Bnd‘𝑋) → 𝑋 ∈ V)
2 elfvex 6696 . . 3 (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
32adantr 481 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑋 ∈ V)
4 fveq2 6663 . . . . . 6 (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5 eqeq1 2822 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
65rexbidv 3294 . . . . . . 7 (𝑦 = 𝑋 → (∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
76raleqbi1dv 3401 . . . . . 6 (𝑦 = 𝑋 → (∀𝑥𝑦𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
84, 7rabeqbidv 3483 . . . . 5 (𝑦 = 𝑋 → {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑥𝑦𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟)} = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)})
9 df-bnd 34938 . . . . 5 Bnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑥𝑦𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟)})
10 fvex 6676 . . . . . 6 (Met‘𝑋) ∈ V
1110rabex 5226 . . . . 5 {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ∈ V
128, 9, 11fvmpt 6761 . . . 4 (𝑋 ∈ V → (Bnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)})
1312eleq2d 2895 . . 3 (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)}))
14 fveq2 6663 . . . . . . . 8 (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
1514oveqd 7162 . . . . . . 7 (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑟) = (𝑥(ball‘𝑀)𝑟))
1615eqeq2d 2829 . . . . . 6 (𝑚 = 𝑀 → (𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1716rexbidv 3294 . . . . 5 (𝑚 = 𝑀 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1817ralbidv 3194 . . . 4 (𝑚 = 𝑀 → (∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1918elrab 3677 . . 3 (𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
2013, 19syl6bb 288 . 2 (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))))
211, 3, 20pm5.21nii 380 1 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cfv 6348  (class class class)co 7145  +crp 12377  Metcmet 20459  ballcbl 20460  Bndcbnd 34926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-bnd 34938
This theorem is referenced by:  bndmet  34940  isbndx  34941  isbnd3  34943  bndss  34945  totbndbnd  34948
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