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Theorem istotbnd 33539
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 6208 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑋 ∈ V)
2 elfvex 6208 . . 3 (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
32adantr 481 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → 𝑋 ∈ V)
4 fveq2 6178 . . . . . 6 (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5 eqeq2 2631 . . . . . . . . 9 (𝑦 = 𝑋 → ( 𝑣 = 𝑦 𝑣 = 𝑋))
6 rexeq 3134 . . . . . . . . . 10 (𝑦 = 𝑋 → (∃𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
76ralbidv 2983 . . . . . . . . 9 (𝑦 = 𝑋 → (∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
85, 7anbi12d 746 . . . . . . . 8 (𝑦 = 𝑋 → (( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
98rexbidv 3048 . . . . . . 7 (𝑦 = 𝑋 → (∃𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
109ralbidv 2983 . . . . . 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 3190 . . . . 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 33538 . . . . 5 TotBnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))})
13 fvex 6188 . . . . . 6 (Met‘𝑋) ∈ V
1413rabex 4804 . . . . 5 {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))} ∈ V
1511, 12, 14fvmpt 6269 . . . 4 (𝑋 ∈ V → (TotBnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))})
1615eleq2d 2685 . . 3 (𝑋 ∈ V → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))}))
17 fveq2 6178 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
1817oveqd 6652 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑑) = (𝑥(ball‘𝑀)𝑑))
1918eqeq2d 2630 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2019rexbidv 3048 . . . . . . . 8 (𝑚 = 𝑀 → (∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2120ralbidv 2983 . . . . . . 7 (𝑚 = 𝑀 → (∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2221anbi2d 739 . . . . . 6 (𝑚 = 𝑀 → (( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2322rexbidv 3048 . . . . 5 (𝑚 = 𝑀 → (∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2423ralbidv 2983 . . . 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 3357 . . 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 276 . 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 368 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 196  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  {crab 2913  Vcvv 3195   cuni 4427  cfv 5876  (class class class)co 6635  Fincfn 7940  +crp 11817  Metcme 19713  ballcbl 19714  TotBndctotbnd 33536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-totbnd 33538
This theorem is referenced by:  istotbnd2  33540  istotbnd3  33541  totbndmet  33542  totbndss  33547  heibor1  33580  heibor  33591
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