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Theorem istotbnd 36637
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 6930 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ 𝑋 ∈ V)
2 elfvex 6930 . . 3 (𝑀 ∈ (Metβ€˜π‘‹) β†’ 𝑋 ∈ V)
32adantr 482 . 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 6892 . . . . . 6 (𝑦 = 𝑋 β†’ (Metβ€˜π‘¦) = (Metβ€˜π‘‹))
5 eqeq2 2745 . . . . . . . . 9 (𝑦 = 𝑋 β†’ (βˆͺ 𝑣 = 𝑦 ↔ βˆͺ 𝑣 = 𝑋))
6 rexeq 3322 . . . . . . . . . 10 (𝑦 = 𝑋 β†’ (βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)))
76ralbidv 3178 . . . . . . . . 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 632 . . . . . . . 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 3179 . . . . . . 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 3178 . . . . . 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 3450 . . . . 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 36636 . . . . 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 6905 . . . . . 6 (Metβ€˜π‘‹) ∈ V
1413rabex 5333 . . . . 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 6999 . . . 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 2820 . . 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 6892 . . . . . . . . . . 11 (π‘š = 𝑀 β†’ (ballβ€˜π‘š) = (ballβ€˜π‘€))
1817oveqd 7426 . . . . . . . . . 10 (π‘š = 𝑀 β†’ (π‘₯(ballβ€˜π‘š)𝑑) = (π‘₯(ballβ€˜π‘€)𝑑))
1918eqeq2d 2744 . . . . . . . . 9 (π‘š = 𝑀 β†’ (𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2019rexbidv 3179 . . . . . . . 8 (π‘š = 𝑀 β†’ (βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2120ralbidv 3178 . . . . . . 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 630 . . . . . 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 3179 . . . . 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 3178 . . . 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 3684 . . 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 287 . 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 380 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  β„+crp 12974  Metcmet 20930  ballcbl 20931  TotBndctotbnd 36634
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-totbnd 36636
This theorem is referenced by:  istotbnd2  36638  istotbnd3  36639  totbndmet  36640  totbndss  36645  heibor1  36678  heibor  36689
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