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Theorem isbnd 37767
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 6945 . 2 (𝑀 ∈ (Bnd‘𝑋) → 𝑋 ∈ V)
2 elfvex 6945 . . 3 (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
32adantr 480 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑋 ∈ V)
4 fveq2 6907 . . . . . 6 (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5 eqeq1 2739 . . . . . . . 8 (𝑦 = 𝑋 → (𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
65rexbidv 3177 . . . . . . 7 (𝑦 = 𝑋 → (∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
76raleqbi1dv 3336 . . . . . 6 (𝑦 = 𝑋 → (∀𝑥𝑦𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
84, 7rabeqbidv 3452 . . . . 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 37766 . . . . 5 Bnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑥𝑦𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟)})
10 fvex 6920 . . . . . 6 (Met‘𝑋) ∈ V
1110rabex 5345 . . . . 5 {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ∈ V
128, 9, 11fvmpt 7016 . . . 4 (𝑋 ∈ V → (Bnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)})
1312eleq2d 2825 . . 3 (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)}))
14 fveq2 6907 . . . . . . . 8 (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
1514oveqd 7448 . . . . . . 7 (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑟) = (𝑥(ball‘𝑀)𝑟))
1615eqeq2d 2746 . . . . . 6 (𝑚 = 𝑀 → (𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1716rexbidv 3177 . . . . 5 (𝑚 = 𝑀 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1817ralbidv 3176 . . . 4 (𝑚 = 𝑀 → (∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
1918elrab 3695 . . 3 (𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
2013, 19bitrdi 287 . 2 (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))))
211, 3, 20pm5.21nii 378 1 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cfv 6563  (class class class)co 7431  +crp 13032  Metcmet 21368  ballcbl 21369  Bndcbnd 37754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-bnd 37766
This theorem is referenced by:  bndmet  37768  isbndx  37769  isbnd3  37771  bndss  37773  totbndbnd  37776
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