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Theorem istotbnd 37776
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 6944 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑋 ∈ V)
2 elfvex 6944 . . 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 6906 . . . . . 6 (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5 eqeq2 2749 . . . . . . . . 9 (𝑦 = 𝑋 → ( 𝑣 = 𝑦 𝑣 = 𝑋))
6 rexeq 3322 . . . . . . . . . 10 (𝑦 = 𝑋 → (∃𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
76ralbidv 3178 . . . . . . . . 9 (𝑦 = 𝑋 → (∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
85, 7anbi12d 632 . . . . . . . 8 (𝑦 = 𝑋 → (( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
98rexbidv 3179 . . . . . . 7 (𝑦 = 𝑋 → (∃𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
109ralbidv 3178 . . . . . 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 3455 . . . . 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 37775 . . . . 5 TotBnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))})
13 fvex 6919 . . . . . 6 (Met‘𝑋) ∈ V
1413rabex 5339 . . . . 5 {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))} ∈ V
1511, 12, 14fvmpt 7016 . . . 4 (𝑋 ∈ V → (TotBnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))})
1615eleq2d 2827 . . 3 (𝑋 ∈ V → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))}))
17 fveq2 6906 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
1817oveqd 7448 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑑) = (𝑥(ball‘𝑀)𝑑))
1918eqeq2d 2748 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2019rexbidv 3179 . . . . . . . 8 (𝑚 = 𝑀 → (∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2120ralbidv 3178 . . . . . . 7 (𝑚 = 𝑀 → (∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2221anbi2d 630 . . . . . 6 (𝑚 = 𝑀 → (( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2322rexbidv 3179 . . . . 5 (𝑚 = 𝑀 → (∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2423ralbidv 3178 . . . 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 3692 . . 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, 25bitrdi 287 . 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 378 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 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480   cuni 4907  cfv 6561  (class class class)co 7431  Fincfn 8985  +crp 13034  Metcmet 21350  ballcbl 21351  TotBndctotbnd 37773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-totbnd 37775
This theorem is referenced by:  istotbnd2  37777  istotbnd3  37778  totbndmet  37779  totbndss  37784  heibor1  37817  heibor  37828
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