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Theorem isbnd 32749
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 6131 . 2 (𝑀 ∈ (Bnd‘𝑋) → 𝑋 ∈ V)
2 elfvex 6131 . . 3 (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
32adantr 480 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑋 ∈ V)
4 fveq2 6103 . . . . . 6 (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5 eqeq1 2614 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
65rexbidv 3034 . . . . . . 7 (𝑦 = 𝑋 → (∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
76raleqbi1dv 3123 . . . . . 6 (𝑦 = 𝑋 → (∀𝑥𝑦𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
84, 7rabeqbidv 3168 . . . . 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 32748 . . . . 5 Bnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑥𝑦𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟)})
10 fvex 6113 . . . . . 6 (Met‘𝑋) ∈ V
1110rabex 4740 . . . . 5 {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ∈ V
128, 9, 11fvmpt 6191 . . . 4 (𝑋 ∈ V → (Bnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)})
1312eleq2d 2673 . . 3 (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)}))
14 fveq2 6103 . . . . . . . 8 (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
1514oveqd 6566 . . . . . . 7 (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑟) = (𝑥(ball‘𝑀)𝑟))
1615eqeq2d 2620 . . . . . 6 (𝑚 = 𝑀 → (𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1716rexbidv 3034 . . . . 5 (𝑚 = 𝑀 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1817ralbidv 2969 . . . 4 (𝑚 = 𝑀 → (∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1918elrab 3331 . . 3 (𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
2013, 19syl6bb 275 . 2 (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))))
211, 3, 20pm5.21nii 367 1 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  cfv 5804  (class class class)co 6549  +crp 11708  Metcme 19553  ballcbl 19554  Bndcbnd 32736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-bnd 32748
This theorem is referenced by:  bndmet  32750  isbndx  32751  isbnd3  32753  bndss  32755  totbndbnd  32758
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