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

Theorem istotbnd 36940
Description: The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
istotbnd (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
Distinct variable groups:   𝑏,𝑑,𝑣,π‘₯,𝑀   𝑋,𝑏,𝑑,𝑣,π‘₯

Proof of Theorem istotbnd
Dummy variables π‘š 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6928 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ 𝑋 ∈ V)
2 elfvex 6928 . . 3 (𝑀 ∈ (Metβ€˜π‘‹) β†’ 𝑋 ∈ V)
32adantr 479 . 2 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))) β†’ 𝑋 ∈ V)
4 fveq2 6890 . . . . . 6 (𝑦 = 𝑋 β†’ (Metβ€˜π‘¦) = (Metβ€˜π‘‹))
5 eqeq2 2742 . . . . . . . . 9 (𝑦 = 𝑋 β†’ (βˆͺ 𝑣 = 𝑦 ↔ βˆͺ 𝑣 = 𝑋))
6 rexeq 3319 . . . . . . . . . 10 (𝑦 = 𝑋 β†’ (βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)))
76ralbidv 3175 . . . . . . . . 9 (𝑦 = 𝑋 β†’ (βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)))
85, 7anbi12d 629 . . . . . . . 8 (𝑦 = 𝑋 β†’ ((βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))))
98rexbidv 3176 . . . . . . 7 (𝑦 = 𝑋 β†’ (βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))))
109ralbidv 3175 . . . . . 6 (𝑦 = 𝑋 β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))))
114, 10rabeqbidv 3447 . . . . 5 (𝑦 = 𝑋 β†’ {π‘š ∈ (Metβ€˜π‘¦) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))} = {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))})
12 df-totbnd 36939 . . . . 5 TotBnd = (𝑦 ∈ V ↦ {π‘š ∈ (Metβ€˜π‘¦) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))})
13 fvex 6903 . . . . . 6 (Metβ€˜π‘‹) ∈ V
1413rabex 5331 . . . . 5 {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))} ∈ V
1511, 12, 14fvmpt 6997 . . . 4 (𝑋 ∈ V β†’ (TotBndβ€˜π‘‹) = {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))})
1615eleq2d 2817 . . 3 (𝑋 ∈ V β†’ (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ 𝑀 ∈ {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))}))
17 fveq2 6890 . . . . . . . . . . 11 (π‘š = 𝑀 β†’ (ballβ€˜π‘š) = (ballβ€˜π‘€))
1817oveqd 7428 . . . . . . . . . 10 (π‘š = 𝑀 β†’ (π‘₯(ballβ€˜π‘š)𝑑) = (π‘₯(ballβ€˜π‘€)𝑑))
1918eqeq2d 2741 . . . . . . . . 9 (π‘š = 𝑀 β†’ (𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2019rexbidv 3176 . . . . . . . 8 (π‘š = 𝑀 β†’ (βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2120ralbidv 3175 . . . . . . 7 (π‘š = 𝑀 β†’ (βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2221anbi2d 627 . . . . . 6 (π‘š = 𝑀 β†’ ((βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2322rexbidv 3176 . . . . 5 (π‘š = 𝑀 β†’ (βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2423ralbidv 3175 . . . 4 (π‘š = 𝑀 β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2524elrab 3682 . . 3 (𝑀 ∈ {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))} ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2616, 25bitrdi 286 . 2 (𝑋 ∈ V β†’ (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))))
271, 3, 26pm5.21nii 377 1 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430  Vcvv 3472  βˆͺ cuni 4907  β€˜cfv 6542  (class class class)co 7411  Fincfn 8941  β„+crp 12978  Metcmet 21130  ballcbl 21131  TotBndctotbnd 36937
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-totbnd 36939
This theorem is referenced by:  istotbnd2  36941  istotbnd3  36942  totbndmet  36943  totbndss  36948  heibor1  36981  heibor  36992
  Copyright terms: Public domain W3C validator