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Theorem istotbnd 32738
Description: The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
istotbnd (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
Distinct variable groups:   𝑏,𝑑,𝑣,𝑥,𝑀   𝑋,𝑏,𝑑,𝑣,𝑥

Proof of Theorem istotbnd
Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6131 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑋 ∈ V)
2 elfvex 6131 . . 3 (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
32adantr 480 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → 𝑋 ∈ V)
4 fveq2 6103 . . . . . 6 (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5 eqeq2 2621 . . . . . . . . 9 (𝑦 = 𝑋 → ( 𝑣 = 𝑦 𝑣 = 𝑋))
6 rexeq 3116 . . . . . . . . . 10 (𝑦 = 𝑋 → (∃𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
76ralbidv 2969 . . . . . . . . 9 (𝑦 = 𝑋 → (∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
85, 7anbi12d 743 . . . . . . . 8 (𝑦 = 𝑋 → (( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
98rexbidv 3034 . . . . . . 7 (𝑦 = 𝑋 → (∃𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
109ralbidv 2969 . . . . . 6 (𝑦 = 𝑋 → (∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
114, 10rabeqbidv 3168 . . . . 5 (𝑦 = 𝑋 → {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))} = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))})
12 df-totbnd 32737 . . . . 5 TotBnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))})
13 fvex 6113 . . . . . 6 (Met‘𝑋) ∈ V
1413rabex 4740 . . . . 5 {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))} ∈ V
1511, 12, 14fvmpt 6191 . . . 4 (𝑋 ∈ V → (TotBnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))})
1615eleq2d 2673 . . 3 (𝑋 ∈ V → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))}))
17 fveq2 6103 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
1817oveqd 6566 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑑) = (𝑥(ball‘𝑀)𝑑))
1918eqeq2d 2620 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2019rexbidv 3034 . . . . . . . 8 (𝑚 = 𝑀 → (∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2120ralbidv 2969 . . . . . . 7 (𝑚 = 𝑀 → (∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2221anbi2d 736 . . . . . 6 (𝑚 = 𝑀 → (( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2322rexbidv 3034 . . . . 5 (𝑚 = 𝑀 → (∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2423ralbidv 2969 . . . 4 (𝑚 = 𝑀 → (∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2524elrab 3331 . . 3 (𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))} ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2616, 25syl6bb 275 . 2 (𝑋 ∈ V → (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))))
271, 3, 26pm5.21nii 367 1 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173   cuni 4372  cfv 5804  (class class class)co 6549  Fincfn 7841  +crp 11708  Metcme 19553  ballcbl 19554  TotBndctotbnd 32735
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-totbnd 32737
This theorem is referenced by:  istotbnd2  32739  istotbnd3  32740  totbndmet  32741  totbndss  32746  heibor1  32779  heibor  32790
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