Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  istotbnd Structured version   Visualization version   GIF version

Theorem istotbnd 35488
 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 6692 . 2 (𝑀 ∈ (TotBnd‘𝑋) → 𝑋 ∈ V)
2 elfvex 6692 . . 3 (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
32adantr 485 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → 𝑋 ∈ V)
4 fveq2 6659 . . . . . 6 (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5 eqeq2 2771 . . . . . . . . 9 (𝑦 = 𝑋 → ( 𝑣 = 𝑦 𝑣 = 𝑋))
6 rexeq 3325 . . . . . . . . . 10 (𝑦 = 𝑋 → (∃𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
76ralbidv 3127 . . . . . . . . 9 (𝑦 = 𝑋 → (∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
85, 7anbi12d 634 . . . . . . . 8 (𝑦 = 𝑋 → (( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
98rexbidv 3222 . . . . . . 7 (𝑦 = 𝑋 → (∃𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
109ralbidv 3127 . . . . . 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 3399 . . . . 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 35487 . . . . 5 TotBnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑦 ∧ ∀𝑏𝑣𝑥𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))})
13 fvex 6672 . . . . . 6 (Met‘𝑋) ∈ V
1413rabex 5203 . . . . 5 {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))} ∈ V
1511, 12, 14fvmpt 6760 . . . 4 (𝑋 ∈ V → (TotBnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))})
1615eleq2d 2838 . . 3 (𝑋 ∈ V → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))}))
17 fveq2 6659 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
1817oveqd 7168 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑑) = (𝑥(ball‘𝑀)𝑑))
1918eqeq2d 2770 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2019rexbidv 3222 . . . . . . . 8 (𝑚 = 𝑀 → (∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2120ralbidv 3127 . . . . . . 7 (𝑚 = 𝑀 → (∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2221anbi2d 632 . . . . . 6 (𝑚 = 𝑀 → (( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2322rexbidv 3222 . . . . 5 (𝑚 = 𝑀 → (∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2423ralbidv 3127 . . . 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 3603 . . 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 290 . 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 384 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 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  {crab 3075  Vcvv 3410  ∪ cuni 4799  ‘cfv 6336  (class class class)co 7151  Fincfn 8528  ℝ+crp 12431  Metcmet 20153  ballcbl 20154  TotBndctotbnd 35485 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7154  df-totbnd 35487 This theorem is referenced by:  istotbnd2  35489  istotbnd3  35490  totbndmet  35491  totbndss  35496  heibor1  35529  heibor  35540
 Copyright terms: Public domain W3C validator